MODELOS DE ISING E POTTS ACOPLADOS AS
TRIANGULAÇÕES DE LORENTZ
José Javier Cerda Hernández
Dissertação/Tese apresentada
ao
Instituto de Matemática e Estatística
da
Universidade de São Paulo
para
obtenção do título
de
Doutor em Ciências
Programa: Estatística
Orientador: Prof. Dr. Anatoli Iambartsev
Coorientador: Prof. Dr. Yuri Suhov
Durante o desenvolvimento deste trabalho o autor recebeu auxílio nanceiro da
CAPES/FAPESP
São Paulo, junho de 2014
MODELOS DE ISING E POTTS ACOPLADOS AS
TRIANGULAÇÕES DE LORENTZ
Esta é a versão original da dissertação/tese elaborada pelo
candidato José Javier Cerda Hernández, tal como
submetida à Comissão Julgadora.
Agradecimentos
First of all I would like to thank my supervisors Anatoli Iambartsev and Yuri Suhov for
guiding me through this research and their professional advisory and patience, as well as for
giving me the freedom to follow dierent themes during my research........
This work was supported by CAPES and FAPESP (projects 2012/04372-7 and 2013/061792). Further, the author thanks the IME at the University of São Paulo for warm hospitality.
..........
i
ii
Resumo
MODELOS DE ISING E POTTS ACOPLADOS
AS TRIANGULAÇÕES DE LORENTZ. 2010. 91 f. Tese (Doutorado) - Instituto de
José Javier Cerda Hernández.
Matemática e Estatística, Universidade de São Paulo, São Paulo, 2010.
O objetivo principal da presente tese é pesquisar : Quais são as propriedades do modelo de
Ising e Potts acoplado ao emsemble de CDT? Para estudar o modelo usamos dois metodos:
(1) Matriz de transferencia e Theorema de Krein-Rutman. (2) Representação FK para o
modelo de Potts sobre CDT e dual de CDT.
Matriz de transferencia permite obter propriedades espectrais da Matriz de transferencia
utilisando o Teorema de Krein-Rutman [KR48] sobre operadores que conservam o cone
de funções positivas. Também obtemos propriedades asintóticas da função de partição e
das medidas de Gibbs. Esses propriedades permitem obter uma região onde a energia livre
converge. O segundo método permite obter uma região onde a curva crítica do modelo
pode estar localizada. Alem disso, também obtemos uma limitante superior e inferior para
a energia livre a volume innito.
Finalmente, utilizando argumentos de dualidade em grafos e expansão em alta temperatura estudamos o modelo de Potts acoplado com triangulações causais. Essa abordagem
permite generalizar o modelo, melhorar os resultados obtidos para o modelo de Ising e obter
novas limitantes, superior e inferior, para a energia livre e para a curva crítica. Alem do
mais, obtemos uma aproximação do autovalor maximal do operador de transferencia a baixa
temperatura.
Palavras-chave:
dinâmica de triangulações causais, modelo de Ising, modelo de Potts,
medida de Gibbs, Teorema de Krein-Rutman, representação FK, modelo de Ising quântico.
iii
iv
Abstract
José Javier Cerda Hernández.
Ising and Potts model coupled to Lorentzian triangu-
lations. 2014. 91 f. Tese (Doutorado) - Instituto de Matemática e Estatística, Universidade
de São Paulo, São Paulo, 2014.
The main objective of the present thesis is to investigate: What are the properties of
the Ising and Potts model coupled to a CDT emsemble? For that objetive, we used two
methods: (1) transfer matrix formalism and Krein-Rutman theory. (2) FK representation of
the
q -state
Potts model on CDTs and dual CDTs.
Transfer matrix formalism permite us obtain spectral properties of the transfer matrix
using the Krein-Rutman theorem [KR48] on operators preserving the cone of positive functions. This yields results on convergence and asymptotic properties of the partition function
and the Gibbs measure and allows us to determine regions in the parameter quarter-plane
where the free energy converges. Second methods permite us determining a region in the
quadrant of parameters
β, µ > 0
where the critical curve for the classical model can be
located. We also provide lower and upper bounds for the innite-volume free energy.
FInally, using arguments of duality on graph theory and hight-T expansion we study
the Potts model coupled to CDTs. This approach permite us improve the results obtained
for Ising model and obtain lower and upper bounds for the critical curve and free energy.
Moreover, we obtain an approximation of the maximal eigenvalue of the transfer matrix at
lower temperature.
Keywords:
causal dynamical triangulation, Ising model, Potts model, Gibbs measure,
Krein-Rutman theory, FK representation, quantum Ising model.
v
vi
Contents
List of Figures
ix
1 Introduction
1
1.1
Introduction and statement results
. . . . . . . . . . . . . . . . . . . . . . .
2 Two-dimensional causal dynamical Triangulations
1
5
2.1
Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Transfer matrix formalism for pure CDTs . . . . . . . . . . . . . . . . . . . .
7
3 Transfer matrix formalism for Ising model coupled to two-dimensional
CDT
13
3.1
The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
The transfer-matrix
. . . . . . . . . . . . . . . . . . .
17
3.3
Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
K
and its powers
KN
4 FK representation for the Ising model coupled to CDT
4.1
The quantum Ising model
4.2
FK representation for Ising model coupled to CDT
4.3
The main results
4.4
Proof of Theorem 4.3.1 and 4.3.2
13
27
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . .
29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
. . . . . . . . . . . . . . . . . . . . . . . .
36
4.4.1
Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.4.2
Proof of Theorem 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5 Potts model coupled to CDTs and FK representation
41
5.1
Introduction and main results of this chapter . . . . . . . . . . . . . . . . . .
41
5.2
Notations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5.2.1
A Potts model coupled to CDTs . . . . . . . . . . . . . . . . . . . . .
45
5.2.2
The FK-Potts model on Lorentzian triangulations . . . . . . . . . . .
46
5.2.3
The relation between the Potts model and FK-Potts model: EdwardsSokal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4
Duality for FK-Potts model coupled to CDTs with periodic boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
47
The proof of Theorem 5.1.1 and rst bounds for the critical curve
vii
. . . . . .
48
52
viii
CONTENTS
5.4
High-T expansion of the Potts model and Proof of Theorem 5.1.2
. . . . . .
58
5.5
Connection between transfer matrix and FK representation . . . . . . . . . .
62
5.5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . .
62
. . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.5.2
5.6
q = 2 (Ising) systems
q -Potts systems . . .
A The von Neumann-Schatten Classes of Operators
Cp
A.1
The space
and rst properties
A.2
The trace class
A.3
The Banach space
A.4
The Hilbert-Schmidt class
C1
. . . . . . . . . . . . . . . . . . . . . . . .
71
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
Cp
72
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
B Krein-Rutman theorem
B.1
Krein-Rutman Theorem and the Principal Eigenvalue . . . . . . . . . . . . .
Bibliography
71
74
75
75
77
List of Figures
S × [j, j + 1].
1.1
A strip of a causal triangulation of
2.1
(a) A strip of a causal triangulation of
. . . . . . . . . . . . . . .
S × [j, j + 1].
2
(b) Geometric represen-
tation of a CDT with periodic spatial boundary condition.
. . . . . . . . . .
7
2.2
Tree parametrization of a causal dynamical triangulation. . . . . . . . . . . .
11
3.1
Illustration of the calculates (3.25) and (3.27). . . . . . . . . . . . . . . . . .
22
3.2
λQ = λ and λT are the maximal eigenvalues of the matrix Q and a related
matrix T respectively. The area above the black curve is where the condition
(3.20) holds true.
4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
ξ = {si }i=1,...,n . Each traarrival time s. In this case,
A trajectory sample associated with a realization
jectory
ϕ ∈ ψξ
can be continuous or not at each
sk−1 the trajectory ϕ do not have jump, and at arrival time
sk the trajectory ϕ have a jump. . . . . . . . . . . . . . . . . . . . . . . . .
In this gure, we show the Cluster Ct of a triangle t, and a graphic represen0
tation of relation t ↔ t , where ↔ on right side in the gure, represent arrival
at arrival time
4.2
times.
4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The area above the minimum of the dotted curve I (graph of the function
30
31
ψ
dened in (4.21)) and dash-dotted line II is where the limiting Gibbs probability measure exists and is unique. The critical curve lies in the region below
the dotted curve I and dash-dotted line II but above the continuous curve III
and dashed line IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Illustrating the region where the critical curve for Potts model coupled CDTs
and dual CDTs can be located.
5.2
t∗
43
with periodic
. . . . . . . . . . . . . . . . . . . . . . . . . . .
47
(a) Geometric representation of a net (b) Geometric representation of a cycle
(c) None of cluster of
5.4
. . . . . . . . . . . . . . . . . . . . . . . . .
Geometric representation of a dual Lorentzian triangulation
spatial boundary condition.
5.3
35
w
is a net or a cycle
Examples of three subgraphs of
ξ(e1 , . . . , e8 )
A
. . . . . . . . . . . . . . . . . . .
49
with 8 edges. It is clear that the term
depends of the topology of the subgraphs.
ix
. . . . . . . . . . . .
59
x
LIST OF FIGURES
5.5
Region where the critical curve of the Ising model coupled to dual CDTs can
be located. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
||A||2 = 1 for q = 4. Black line: µ = 3 ln 2.
∗
− 1 + ln 2. Red line: µ∗ = 32 ln 42/3 + eβ − 1 + ln 2.
The blue line is the simulation of
Green line:
β∗
µ∗ = 23 ln e
63
∗
68
Chapter 1
Introduction
1.1 Introduction and statement results
Models of planar random geometry appear in physics in the context of two-dimensional
quantum gravity and provide an interplay between mathematical physics and probability
theory.
Causal dynamical triangulation (CDT), introduced by Ambjørn and Loll (see [AL98]),
together with its predecessor a dynamical triangulation (DT), constitute attemps to provide
a meaning to formal expressions appearing in the path integral quantisation of gravity (see
[ADJ97], [AJ06] for an overview). A causal triangulation is formed by triangulations of spatial strips as illustrated in Figure 5.2. Note that the left and right boundaries of the spatial
strip are periodically identied. The idea is to regularise the path integral by approximating
the geometries emerging in the integration by CDTs. As a result, the path integral over
geometries is replaced with a sum over all possible triangulations where each conguration
is weighted by a Boltzmann factor
tion and
µ
e−µ|T | ,
with
|T |
standing for the size of the triangula-
being the cosmological constant. The evaluation of the partition function was
reduced to a purely combinatorial problem that can be solved with the help of the early
work of Tutte [Tut62, Tut63]; alternatively, more powerful techniques were proposed, based
on random matrix models (see, e.g., [FGZJ95]) and bijections to well-labelled trees (see
[Sch97, BDG02]).
From a probabilistic point of view there has recently been an increasing interest in DT,
most notably through the work of Angel and Schramm on a uniform measure on innite
planar triangulations [AS03], as well as through the work of Le Gall, Miermont and collaborators on Brownian maps (see [GG11] for a recent review).
From a physical point of view it is interesting to study various models of matter, such
as the Ising and Potts model, coupled to the CDT. An interesting question is: What are
the properties of the Ising and Potts model coupled to a CDT ensemble? It is still random
and allows for a back-reaction of the spin system with the quantum geometry. Monte Carlo
simulations [AAL99] (see also [BL07, AALP08]) give a strong evidence that critical exponents
1
2
1.1
INTRODUCTION
S ×[ j, j +1]
down"
up"
Figure 1.1:
root"
A strip of a causal triangulation of S × [j, j + 1].
of the Ising model coupled to CDT are identical to the Onsager values. The calculation of the
partition function in this case also reduces to a combinatorial problem. It was rst solved in
[Kaz86, BK87] by using random matrix models and later by using a bijection to well-labelled
trees [BMS11]. It is interesting that the solution here is much simpler than in the case of
a at triangular or square lattice as given by Onsager [Ons44]. For the
2-state
Potts model
(Ising model) coupled to CDTS some progress has been recently made on existence of Gibbs
measures and phase transitions (see [AAL99], [BL07], [HYSZ13] and [Her14] for details).
Using transfer matrix methods, the Krein-Rutman theory of positivity-preserving operators
and FK representation for the Ising model, [Her14] provides a region in the quadrant of
parameters
β, µ > 0
where the innite-volume free energy has a limit, providing results on
convergence and asymptotic properties of the partition function and the Gibbs measure.
Thus, FK-Potts models, introduced by Fortuin and Kasteleyn (see [FK72]), prove that these
models have become an important tool in the study of phase transition for the Ising and
q -state
Potts model.
The goal of this thesis is to use Krein-Rutman theory of positivity-preserving operators,
FK representation of the q -state Potts model on a xed triangulation and duality theory of
graph for study the q -state Potts model coupled to CDTs.
While recently much progress has been made in the development of analytical techniques
for CDT [JAZ07, JAZ08d], particularly random matrix models [JAZ08b, JAZ08a, JAZ08c],
and their application to multi-critical CDT [AGGS12, AZ12a, AZ12b], the causality constraints still makes it dicult to nd an analytical solution of the Ising model coupled to
CDT.
In this thesis we focus on study the
q -state Potts model coupled to CDTs and is organised
as follows.
In Chapter 2 gives a summary of causal dynamical triangulations CDTs and we intro-
1.1
INTRODUCTION AND STATEMENT RESULTS
3
duced the transfer matrix formalism for pure CDTs. Also, we study asymptotic properties
of the partition function for pure CDTs. These properties will be used in next chapters.
In Chapter 3 we dene the annealed Ising model coupled to two-dimensional CDT and
develop a transfer matrix formalism. Spectral properties of the transfer matrix are rigorously
analysed by using the Krein-Rutman theorem [KR48] on operators preserving the cone
of positive functions. This yields results on convergence and asymptotic properties of the
partition function and the Gibbs measure and allows us to determine regions in the parameter
quarter-plane where the partition function converges. The main results of this chapter are
Lemma 3.2.1 and Theorem 3.2.2.
In Chapter 4 we use the Fortuin-Kasteleyn (FK) representation of quantum Ising models
via path integrals for determining a region in the quadrant of parameters
β, µ > 0
where
the critical curve for the classical model can be located. In Section 4.1 we describe the
quantum Ising model. In Section 4.2, we give the FK representation of Ising model coupled
to CDTs via a path integral. This representation was originally derived in [MAC92] (see also
[Aiz94] and [Iof09]). Section 4.3 we present the main results of this chapter (Theorems 4.3.1
and 4.3.2). Section 4.4.1 and 4.4.2 contains the proof of Theorems 4.3.1 and 4.3.2. We also
provide lower and upper bounds for the innite-volume free energy. This chapter extends
results from Chapter 3 for the (annealed) Ising model coupled to two-dimensional causal
dynamical triangulations.
In Chapter 5. In Section 5.2, we introduce notation, dened the Potts model coupled
to CDTs and give a summary of the FK model, FK representation. Finally, we establish a
technical proposition of duality that will used in the next section. Section 5.3 contains the
proof of the rts main Theorem 5.1.1, and we nd a rst bounds for the critical curve. This
result will play a key role proof of the second main Theorem 5.1.2 of this chapter. In Section
5.4, using the High-T expansion for
q -state
Potts model, we prove Theorem 5.1.2.
Finally, Appendix A and B provide a review of trace class operators and Krein-Rutman
theory, used in Chapters 2 and 3.
Most of the novel results of this thesis have been published in research articles. In particular, the following chapters are based on the following articles:
•
Chapter 2 and 3 on J.C. Hernández, Y. Suhov, A. Yambartsev, and S. Zohren, Bounds
on the critical line via transfer matrix methods for an Ising model coupled to causal
dynamical triangulations. J. Math. Phys.
•
54 063301 (2013).
Chapter 4 on submitted paper, J. Cerda-Hernández, Critical region for an Ising model
coupled to causal dynamical triangulations. arxiv 1402.3251 (2014).
•
Chapter 5 on preparation article, J. Cerda-Hernández, Duality relation for Potts model
coupled to causal dynamical triangulations (2014).
4
INTRODUCTION
1.1
Chapter 2
Two-dimensional causal dynamical
Triangulations
In this chapter we introduce causal dynamical triangulations (CDTs) as a discretization
of the partition function for two-dimensional quantum gravity. After giving a mathematical
denition of CDT we show some asymptotical properties of the partition function using
transfer matrix approach. These asymptotical properties will used in next sections.
2.1 Denitions
We will work with rooted causal dynamic triangulations of the cylinder
N = 1, 2, . . . , which have N
bonds (strips)
S × [j, j + 1]. Here S
CN = S × [0, N ],
stands for a unit circle. The
denition of a causal triangulation starts by considering a connected graph
in
CN
with the property that all faces of
G
G
embedded
are triangles (using the convention that an
edge incident to the same face on two sides counts twice, see [SYZ13] for more details). A
triangulation
t
of
CN
is a pair formed by a graph
of all its (triangular) faces:
G
with the above propetry and the set
F
t = (G, F ).
Denition 2.1.1.
A triangulation t of CN is called a causal triangulation (CT) if the
following conditions hold:
• each triangular face of t belongs to some strip S × [j, j + 1], j = 1, . . . , N − 1, and has
all vertices and exactly one edge on the boundary (S × {j}) ∪ (S × {j + 1}) of the strip
S × [j, j + 1];
• if kj = kj (t) is the number of edges on S × {j}, then we have 0 < kj < ∞ for all
j = 0, 1, . . . , N − 1.
Denition 2.1.2.
A triangulation t of CN is called rooted if it has a root. The root in the
triangulation t is represented by a triangular face t of t, called the root triangle, with an
anticlock-wise ordering on its vertices (x, y, z) where x and y belong to S 1 × {0}. The vertex
x is identied as the root vertex and the (directed) edge from x to y as the root edge.
5
6
2.1
TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS
Denition 2.1.3. Two causal rooted triangulations of CN , say t = (G, F ) and t0 = (G0 , F 0 ),
are equivalent if there exists a self-homeomorphism of CN which (i) transforms each slice
S 1 × {j}, j = 0, . . . , N − 1 to itself and preserves its direction, (ii) induces an isomorphism
of the graphs G and G0 and a bijection between F and F 0 , and (iii) takes the root of t to the
root of t0 .
LT∞ denote the sets of causal triangulations on the nite cylinder CN
innity cylinder C = S × [0, ∞).
A triangulation t of CN is identied as a consistent sequence:
Let
LTN
and
and
t = (t(0), t(1), . . . , t(N − 1)),
where
t(i)
is a causal triangulation of the strip
S × [i, i + 1].
The latter means that each
S × [i, i + 1] into triangles where each triangle has one
vertex on one of the slices S × {i}, S × {i + 1} and two on the other, together with the edge
joining these two vertices. The property of consistency means that each pair (t(i), t(i + 1))
is consistent, i.e., every side of a triangle from t(i) lying in S × {i + 1} serves as a side of a
triangle from t(i + 1), and vice versa.
The triangles forming the causal triangulation t(i) are denoted by t(i, j), 1 ≤ j ≤ n(t(i))
where, n(t(i)) stands for the number of triangles in the triangulation t(i). The enumeration
of these triangles starts with what we call the root triangle in t(i); it is determined recursively
as follows (see Figure 2.1(b)): First, we have the root triangle t(0, 1) in t(0) (see Denition
2.1.2). Take the vertex of the triangle t(0, 1) which lies on the slice S × {1} and denote it
0
by x . This vertex is declared the root vertex for t(1). Next, the root edge for t(1) is the one
0
0
0
incident to x and lying on S ×{1}, so that if y is its other end and z is the third vertex of the
0 0 0
corresponding triangle then x , y , z lists the three vertices anticlock-wise. Accordingly, the
0 0 0
triangle with the vertices x , y , z is called the root triangle for t(1). This construction can be
iterated, determining the root vertices, root edges and root triangles for t(i), 0 ≤ i ≤ N − 1.
t(i)
is described by a partition of
It is convenient to introduce the notion of up" and down" triangles (see Figure 2.1(a)).
We call a triangle
t ∈ t(i)
an up-triangle if it has an edge on the slice
S × {i}
and a down-
S ×{i+1}. By Denition 2.1.1, every triangle is either of
type up or down. Let nup (t(i)) and ndo (t(i)) stand for the number of up- and down-triangles
in the triangulation t(i).
Note that for any edge lying on the slice S × {i} belongs to exactly two triangles: one uptriangle from t(i) and one down-triangle from t(i − 1). This provides the following relation:
the number of triangles in the triangulation t is twice the total number of edges on the slices.
i
More precisely, let n be the number of edges on slice S×{i}. Then, for any i = 0, 1, . . . , N −1,
triangle if it has an edge on the slice
n(t(i)) = nup (t(i)) + ndo (t(i)) = ni + ni+1 ,
(2.1)
2.2
TRANSFER MATRIX FORMALISM FOR PURE CDTS
7
S1 ×[ i, i +1]
down
triangle
up
triangle
root
(a)
(b)
Figure 2.1: (a) A strip of a causal triangulation of S × [j, j + 1]. (b) Geometric representation of
a CDT with periodic spatial boundary condition.
implying that
N
−1
X
n(t(i)) = 2
i=0
N
−1
X
ni .
(2.2)
i=0
There is another useful property regarding the counting of triangulations. Let us x the
number of edges
ni
ni+1 in the slices S × {i} and S × {i + 1}. The number of possible
S × [i, i + 1] with ni up- and ni+1 down-triangles is equal to
and
rooted CTs of the slice
ni + ni+1 − 1
n(t(i)) − 1
=
.
ni − 1
nup (t(i)) − 1
(2.3)
2.2 Transfer matrix formalism for pure CDTs
We begin by discussing the case of pure causal dynamical triangulations, as was rst
introduced in [AL98] (see also [MYZ01] for a mathematically more rigorous account).
The partition function for rooted CTs in the cylinder
conditions (where
constant
µ
t(0)
is consistent with
t(N − 1))
CN
with periodical spatial boundary
and for the value of the cosmological
is given by
ZN (µ) =
X
−µn(t)
e
t
=
X
N
−1
n
o
X
exp −µ
n(t(i)) .
(t(0),...,t(N −1))
i=0
(2.4)
Using the properties (2.2) and (2.3) we can represent the partition function (2.4) in the
following way
ZN (µ) =
X
n0 ≥1,...,nN −1 ≥1
N
−1
−1 i
n
o NY
X
n + ni+1 − 1
i
exp −2µ
n
.
i−1
n
i=0
i=0
(2.5)
8
2.2
TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS
Moreover,
ZN (µ)
admits a trace-related representation
ZN (µ) = tr U N .
This gives rise to a transfer matrix
(2.6)
U = {u(n, n0 )}n,n0 =1,2,...
describing the transition from
one spatial strip to the next one. It is an innite matrix with strictly positive entries
n + n0 − 1 n+n0
u(n, n ) =
g
.
n−1
0
For notational convenience we use the parameter
g = e−µ
(2.7)
(a single-triangle fugacity). The
0
u(n, n ) yields the number of possible triangluations of a single strip (say, S × [0, 1])
0
with n lower boundary edges (on S × {0}) and n upper boundary edges (on S × {1}). See
0
Figure 5.2. The asymmetry in n and n is due to the fact that the lower boundary is marked
e = {e
while the upper one is not. However, a symmetric transfer matrix U
u(n, n0 )} can be
entry
introduced, associated with a strip where both boundaries are kept unmarked:
u
e(n, n0 ) = n−1 u(n, n0 ).
(2.8)
N -strip Gibbs distribution PN assigns the following probabilities to strings (n0 , . . . , nN −1 )
i
the number of triangles n ≥ 1 for all i = 0, . . . , N − 1:
The
with
0
PN,µ (n , . . . , n
N −1
N
−1
−1 i
n
o NY
X
n + ni+1 − 1
1
i
exp −2µ
n
.
)=
i−1
ZN (µ)
n
i=0
i=0
We state two lemmas featuring properties of matrix
(2.9)
U:
Lemma 2.2.1.
For any g > 0 the matrix U and its transpose U T have an eigenvalue
Λ = Λ(g) given by
h
i2
p
Λ(g) = (1 − 1 − 4g 2 )/(2g) .
(2.10)
The corresponding eigenvectors
φ = {φ(n)}n=1,2,... and φ∗ = {φ∗ (n)}n=1,2,...
have entries
n
φ(n) = n Λ(g) , φ∗ (n) = (Λ(g))n .
Proof.
(2.11)
A direct verication shows that
X
n0
0
u(n, n0 )n0 Λn (g) = nΛn+1 (g)
and
X
0
Λn (g)u(n, n0 ) = Λn +1 (g).
n
(In fact, each of these relations implies the other.) See Theorem 1 in [MYZ01].
2.2
TRANSFER MATRIX FORMALISM FOR PURE CDTS
Lemma 2.2.2.
For any xed n and any g < 1 (equivalently, µ > 0) one has
X
u(n, n0 ) =
n0
Proof.
9
g n
1 − (1 − g)n .
1−g
(2.12)
The proof again follows from a straightforward verication.
A transfer-matrix formalism of Statistical Mechanics predicts that, as
partition function is governed by the largest eigenvalue
Λ
N → ∞,
the
of the transfer matrix:
ZN (g) = tr U N ∼ ΛN
(2.13)
We make this statement more precise in the statements of Lemma 2.2.3 and Theorem 2.2.1
below. Here the symbol
`2
stands for the Hilbert space of square-summable complex se-
quences (innite-dimensional vectors) ψ = {ψ(n)}n=1,2,... equipped with the standard scalar
P 0
00
T
0
00
product hψ , ψ i =
are treated as opn ψ (n)ψ (n). Accordingly, the matrices U and U
2
erators in ` .
Lemma 2.2.3.
For any g < 1/2 (equivalently µ > ln 2) the following statements hold true:
1. U and U T are bounded operators in `2 preserving the cone of positive vectors;
2. The sum
P
n,n0
u(n, n0 ) < ∞. Consequently, U and U T have
tr U U T = tr U T U < ∞,
i.e., U and U T are Hilbert-Schmidt operators. Therefore, ∀ N ≥ 2, U N and U T
trace-class operators.
N
are
3. The maximal eigenvalue Λ = Λ(g) of U in `2 is positive, coincides with the maximal
eigenvalue of U T and is given by Eqn (2.10). The corresponding eigenvectors φ, φ∗ ∈ `2
are unique up to multiplication by a constant factor and given in Eqn (2.11).
4. The following asymptotical formulas hold as N → ∞:
1
1
N
T N
tr
U
,
tr
(U
)
→ 1,
ΛN
ΛN
and, ∀ vectors ψ 0 , ψ 00 ∈ `2 ,
1
hψ 0 , U N ψ 00 i = hψ 0 , φihφ∗ , ψ 00 i,
N
Λ
where the eigenvectors φ and φ∗ are normalized so that hφ, φ∗ i = 1.
Theorem 2.2.1.
For any g < 1/2 the following relation holds true:
1
log ZN (g) = log Λ
N →∞ N
lim
(2.14)
10
2.2
TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS
with Λ = Λ(g) given in (2.10). Further, the N -strip Gibbs measure PN,µ converges weakly to
a limiting measure Pµ which is represented by a positive recurrent Markov chain on Z+ =
{1, 2, . . .}, with the transition matrix P = {P (n, n0 )}n=1,2,... and the invariant distribution π .
Here
u(n, n0 )φ(n0 )
P (n, n0 ) =
Λφ(n)
and
π(n) =
φ∗ (n)φ(n)
.
hφ∗ , φi
where φ(n) and φ∗ (n) are as in (2.11).
Proof.
The proof is a consequence of Lemma 2.2.1 and 2.2.3 and the Krein-Rutman theory
[KR48].
By Theorem 2.2.1, the measure on the set of innite triangulations
LT∞
is then dened
as a weak limit
Pµ = lim PN .
N →∞
The follow Theorem given the typical triangulation (typical behavior) under the limiting
measure
Pµ .
Theorem 2.2.2
.
The limit measure Pµ = limN →∞ PN,µ exist for
all µ ≥ ln 2. Moreover, let nk be the number of vertices at k -th level in a triangulation t for
each k ≥ 0.
(See [MYZ01], [KY12])
• For µ > ln 2 under the limiting measure Pµ the sequence {nk } is a positive recurrent
Markov chain.
• For µ = µcr = ln 2 the sequence {nk } is distributed as the branching process ξn with
geometric ospring distribution with parameter 1/2, conditioned to non-extinction at
innity.
Below we briey sketch the proof of the second part of Theorem 2.2.2, a deeper investigation of related ideas will appear in [SYZ13].
Given a triangulation
v∈t
t
t
v
v
τ ⊂ t
by taking, for each vertex
downwards (see g. 1). The graph thus obtained is a
, and moreover, if one associates with each vertex of
can be completely reconstructed knowing
For every vertex
from
dene the subgraph
, the leftmost edge going from
spanning forest of
then
t ∈ LTN ,
v∈τ
denote by
δv
τ.
We call
τ
τ
it is height in
the tree parametrization of
it is out-degree, i.e. the number of edges of
upwards. Comparing the out-degrees in
τ
τ
n(t),
t.
going
to the number of vertical edges in
and comparing the latter to the total number of triangles
t
t
,
it is not hard to obtain the
identity
X
(δv + 1) = n(t),
v∈τ \S×{N }
(2.15)
2.2
TRANSFER MATRIX FORMALISM FOR PURE CDTS
Figure 2.2:
Tree parametrization of a causal dynamical triangulation.
where the sum on the left runs over all vertices of
the measure
Pµcr
11
the probability of a forest
−µcr n(t)
e
τ
v∈τ \S×{N }
which is exactly the probability to observe
τ
except for the
N -th
level. Thus, under
is proportional to
Y
=
τ
δv +1
1
,
2
(2.16)
as a realization of a branching process with
ospring distribution Geom(1/2). After normalization we will obtain, on the left in (2.16),
PN,µcr (τ ) as dened by (2.9), an on the right the conditional probability to see
τ as a realization of the branching process ξ given ξN > 0. So quite naturally when N → ∞
the distribution of τ converges to the Galton-Watson tree, conditioned to non-extinction at
the probability
innity.
In particular it follows from Theorem 2.2.2 that
Pµcr (nk = m) = P r(ξk = m|ξ∞ > 0) = mP r(ξk = m)
(2.17)
Remark 2.2.1.
The last equality in (2.17) means that the measure Pµcr on triangulations
can be considered as a Q-process dened by Athreya and Ney [AN72] for a critical GaltonWatson branching process. Such a process is exactly a critical Galton-Watson tree conditioned
to survive forever.
In the supercritical case
exp(−µ) < 1/2,
we have the following asymptotical property of
the partition function
Proposition 2.2.1.
In the supercritical case, exp(−µ) < 1/2, the nite volume partition
function ZN (µ) (dened in (2.4)) exist only if
π
µ > ln 2 cos
N +1
Notice that, as
N →∞
.
(2.18)
this region, where the partition function exists, become empty.
12
TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS
Remark 2.2.2.
2.2
The inequality in (2.18) means that if µ < ln 2 then there exists N0 ∈ N
such that the partition function ZN (µ) = +∞ whenever N > N0 . Moreover, the Gibbs
distribution PN,µ on triangulations with periodic boundary conditions cannot be dened by
using the standard formula with PN,µ as a normalising denominator, consequently, there is
no limiting probability measure Pµ .
Chapter 3
Transfer matrix formalism for Ising
model coupled to two-dimensional CDT
In this chapter we introduce a transfer matrix formalism for the (annealed) Ising model
coupled two-dimensional CDTs. Using the Krein-Rutman theory of positivity preserving
operators we study several properties of the emerging transfer matrix. In particular, we
determine regions in the quadrant of parameters
β, µ > 0
where the innite-volume free
energy converges, yields results on the convergence and asymptotic properties of the partition
function and Gibbs measure. This is a rst approach for study the Ising model coupled twodimensional CDTs.
3.1 The model
t = (t(0), t(1), . . . , t(N − 1)) be a triangulation of CN , where t(i) is a causal triangulation of the strip S × [i, i + 1]. The triangles forming the causal triangulation t(i) are
denoted by t(i, j), 1 ≤ j ≤ n(t(i)) where, n(t(i)) stands for the number of triangles in
the triangulation t(i). The enumeration of these triangles starts with what we call the root
triangle in t(i) (see Chapter 2).
Now, with any triangle from a triangulation t we associate a spin taking values ±1. An
N -strip conguration of spins is represented by a collection
Let
σ = (σ(0), σ(1), . . . , σ(N − 1))
σ(i) = σ(t(i)) is a conguration of spins σ(i, j) over triangles t(i, j) forming a triangulation t(i), 1 ≤ j ≤ n(t(i)). We will say that a single-strip conguration of spins σ(i) is
supported by a triangulation t(i) of strip S × [i, i + 1]. We consider a usual (ferromagnetic)
0 0
Ising-type energy where two spins σ(i, j) and σ(i , j ) interact if their supporting triangles
t(i, j), t(i0 , j 0 ) share a common edge; such triangles are called nearest neighbors, and this
0 0
0
property is reected in the notation hσ(i, j), σ(i , j )i, where we require 0 ≤ i ≤ i ≤ N − 1.
0 0
Thus, in our model each spin has three neighbors. Moreover, a pair hσ(i, j), σ(i , j )i can
where
13
14
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.1
TWO-DIMENSIONAL CDT
only occur for
i0 − i ≤ 1
i = 0, i0 = N − 1.
or
Formally, the Hamiltonian of the model reads:
X
H(σ) = −
σ(i, j)σ(i0 , j 0 ).
(3.1)
hσ(i,j),σ(i0 ,j 0 )i
We will use the following decomposition:
H(σ) =
N
−1
X
H(σ(i)) +
i=0
where we assume that
represents the energy
N
−1
X
V (σ(i), σ(i + 1)),
(3.2)
i=0
σ(0) ≡ σ(N ) (the periodic spatial boundary condition). Here H(σ(i))
of the conguration σ(i):
X
H(σ(i)) = −
σ(i, j)σ(i, j 0 ).
(3.3)
hσ(i,j),σ(i,j 0 )i
V (σ(i), σ(i+1)) is the energy of interaction between neighboring triangles belonging
adjacent strips S × [i, i + 1] and S × [i + 1, i + 2]:
Further,
to the
X
V (σ(i), σ(i + 1)) = −
σ(i, j)σ(i + 1, j 0 ).
(3.4)
hσ(i,j),σ(i+1,j 0 )i
The partition function for the (annealed)
inverse temperature
β>0
Ising model coupled to CDT, at the
and for the cosmological constant
X
N
−1
n
o
X
exp −µ
n(t(i))
(t(0),...,t(N −1))
i=0
ΞN (µ, β) =
X
×
N -strip
N
−1
Y
µ,
is given by
(3.5)
n
o
exp −βH(σ(i)) − βV (σ(i), σ(i + 1)) .
(σ (0),...,σ (N −1)) i=0
Here
n(t(i))
stands for the number of triangles in the triangulation
t(i).
Like before, the
formula
ΞN (µ, β) = tr KN
(3.6)
K with entries K((t, σ), (t0 , σ 0 )) labelled by pairs (t, σ), (t0 , σ 0 )
of a single strip (say, S × [0, 1]) and their supported spin cong-
gives rise to a transfer matrix
representing triangulations
urations which are positioned next to each other. Formally,
o
n µ
0
K((t, σ), (t , σ )) = 1t∼t0 exp − (n(t) + n(t ))
2
n β
o
× exp − H(σ) + H(σ 0 ) − βV (σ, σ 0 ) .
2
0
As earlier,
indicator
n(t)
1t∼t0
and
n(t0 )
0
are the numbers of triangles in the triangulations
means that the triangulations
0
t, t
(3.7)
t
and
t0 .
The
have to be consistent with each other in the
3.1
THE MODEL
above sense: the number of down-triangles in
t
15
should equal the number of up-triangles in
t0 , and an upper-marked edge in t should coincide with a lower-marked edge in triangulation
t0 . It means that the pair (t, t0 ) forms a CDT for the strip S × [0, 2].
N
We would like to stress that the trace tr K
in (3.6) is understood as the matrix trace,
P
(N )
i.e., as the sum
((t, σ), (t, σ)) of the diagonal entries K (N ) ((t, σ), (t, σ)) of the
t,σ K
N
matrix K . (Indeed, in what follows, the notation tr is used for the matrix trace only.)
Our aim will be to verify that the matrix trace in (3.6) can be replaced with an
trace
invoking the eigenvalues of
K
As before, we can introduce the
operator
in a suitable linear space (see next section).
N -strip
Gibbs probability distribution associated with
formula (3.5):
PN (t(0), σ(0)), . . . , (t(N − 1), σ(N − 1))
N
−1
n
o
Y
1
exp −µn(t(i)) − βH(σ(i)) − βV (σ(i), σ(i + 1)) .
=
Ξ(µ, β) i=0
(3.8)
Consider several special cases of interest.
The case
β ≈ 0. This is the rst term of the so-called high temperature expansion [AAL99].
Here one has
X
N
−1
n
o
X
exp −µ
n(t(i))
X
(t(0),...,t(N −1))
i=0
(σ (0),...,σ (N −1))
Ξ(µ, 0) =
n
exp −2(µ − ln 2)
X
=
n0 ≥1,...,nN −1 ≥1
= ZN (µ − ln 2);
The condition
µ − ln 2 > ln 2
N
−1
X
i
n
i=0
1
−1 i
o NY
n + ni+1 − 1
i=0
ni − 1
cf. (2.4).
which guarantees properties listed in Lemma 2.2.3 and
Theorem 2.2.1 resuls in
µ > 2 ln 2.
Thus, Eqn. (3.9) yields a sub-criticality condition when
(3.9)
β = 0.
β ≈ ∞. Observe that for any triangulation t = (t(0), . . . , t(N − 1)) there are two
ground states: all spins +1 and all spins −1, with the overall energy equals minus three
PN −1
half times the total number of triangles: −3/2
i=0 n(t(i)). Discarding all other spin
The case
congurations, we obtain that
Ξ(µ, β) > Ξ∗ (µ, β)
16
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.1
TWO-DIMENSIONAL CDT
where
X
Ξ∗ (µ, β) =
t(0),...,t(N −1)
N −1
o
n
3 X
n(t(i))
2 exp −µ + β
2
i=0
X
= 2
n0 ≥1,...,nN −1 ≥1
N −1
n
3 X i o ni + ni+1 − 1
n
exp −2 µ − β
2
ni − 1
i=0
3
= 2ZN µ − β
2
#
3 X
β
n(t(i)) is the energy of the (+)-conguration (or, equivalently,
where exp
2 i
the (−)-conguration). For β large, we can expect that Ξ(µ, β) ∼ Ξ∗ (µ, β). Then the
"
critical inequality
3
µ − β > ln 2
2
yields
3
µ > ln 2 + β.
2
(3.10)
Equation (3.10) gives a necessary (and probably tight) criticality condition for the
Ising model under consideration for large values of
β.
A similar result was obtained in
[AAL99].
The case
0 < β < ∞.
Firstly, we note that for any xed triangulation
spin conguration
(all
+s
or all
t
the energy of any
σ on t will be bigger or equal than the energy of a pure conguration
−s):
H(σ) =
X
H(σ(j)) +
X
j
3
≥ − #(of
2
V (σ(j), σ(j + 1))
j
all triangles in
t) = −3
N
−1
X
ni ,
i=0
ni is the number of edges in the ith level S × {i}, i = 0, 1 . . . , N − 1.
β > 0 the inequality Ξ(µ, β) < Ξ∗ (µ, β) holds true, where
where
any
∗
Ξ (µ, β) =
X
exp
(t(0),...,t(N −1)
=
X
n0 ≥1,...,nN −1 ≥1
n
−1
o
X
N
3
−µ + β + ln 2
n(t(i))
2
i=0
−1
n
o
X
N
3
ni
exp −2 µ − β − ln 2
2
i=0
3
= ZN µ − β − ln 2
2
.
Thus, for
3.2
THE TRANSFER-MATRIX
K
AND ITS POWERS
KN
17
Hence, the inequality
3
µ − β − ln 2 > ln 2
2
3
µ > 2 ln 2 + β
2
or
(3.11)
provides a sucient condition for subcriticality of the Ising model under consideration.
3.2 The transfer-matrix K and its powers KN
The main results of this chapter are summarized in Lemma 3.2.1 and Theorems 3.2.1
and 3.2.2 below.
Let us start with a statement (see Proposition 3.2.1 below) which merely re-phrases
standard denitions and explains our interest in the matrices
K, KT , KT K, KKT
and their
powers. Cf. Denition 2.2.2 on p.83, Denition 2.4.1 on p.101, Lemma 2.3.1 on p.85 and
Theorem 3.3.13 on p.139 in [Rin71]). See Appendix A for a short review.
We treat the transfer-matrix
K
and its transpose
KT
as linear operators in the Hilbert
2
space `T−C (the subscript T-C refers to triangulations and spin-congurations). The space
`2T−C is formed by functions ψ = {ψ(t, σ)} with the argument (t, σ) running over single-strip
0
00
triangulations and supported congurations of spins, with the scalar product hψ , ψ iT−C =
P
0
00
2
t,σ ψ (t, σ)ψ (t, σ) and the induced norm kψkT−C . The action of K in `T−C , in the basis
formed by Dirac's delta-vectors
δ(t,σ ) ,
is determined by
X
Kψ (t, σ) =
K((t, σ), (t0 , σ 0 ))ψ(t0 , σ 0 );
t 0 ,σ 0
(3.12)
K, KT , etc., for the matrices and the corresponding operators
`2T−C . Accordingly, the symbols kKkT−C , kKT kT−C etc. refer to norms in `2T−C .
n
T n
Given n = 1, 2, . . ., suppose that the operator K (respectively, K
) is of trace class
in following we use the notation
in
(see denition in Appendix A). Then the following series absolutely converges:
!
X
(n)
Λj
respectively,
j
where
(n)
Λj
X
(n)
Λ∗ j
,
(3.13)
j
∗ (n)
(Λ j ) runs through the eigenvalues of
Kn ((KT )n ),
counted with their multi-
plicities. In this case the sum (3.13) is called the operator trace of
in
`2T−C .
(respectively,
(KT )n )
We adopt an agreement that the eigenvalues in (3.13) are listed in the decreasing
order of their moduli, beginning with
Set
Kn
|Kn | =
p
(KT )n Kn
and
(n)
Λ0
∗ (n)
(Λ 0 ).
T n p
K = Kn (KT )n .
18
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.2
TWO-DIMENSIONAL CDT
Proposition 3.2.1.
For any positve integer r, the following inequalities are equivalent:
tr
Kr (KT
r = tr
KT
r
Kr < ∞ and
(3.14)
tr|K2r | = tr|(KT )2r | < ∞.
Moreover, each of the inequalities in (3.14) implies that ∀ N ≥ 2r, the operators KN
and (KT )N are of trace class in `2T−C . Hence, for N ≥ 2r, the matrix traces tr KN and
tr((KT )N ) are nite and coincide with the corresponding operator traces in `2T−C .
Theorem 3.2.1. Suppose that the condition (3.14) is satised with r = 1. Then the following
properties of transfer matrix K are fulllled.
1. The square K2 and its transpose (KT )2 are trace-class operators in `2T−C .
2. K and KT have a common eigenvalue, Λ = Λ0 (β, µ) > 0 such that the norms
kKkT−C = kKT kT−C = Λ. Furthermore, K2 and (KT )2 have the common eigenvalue
(2)
(2)
Λ2 = Λ0 = Λ∗ 0 such that the norms kK2 kT−C = k(KT )2 kT−C = Λ2 .
3. Λ is a simple eigenvalue of K and KT , i.e., the corresponding eigenvectors φ =
{φ(t, σ)} and φ∗ = {φ∗ (t, σ)} are unique up to multiplicative constants. Moreover,
φ and φT can be made strictly positive: φ(t, σ), φT (t, σ) > 0 ∀ (t, σ). Furthermore,
Λ is separated from the remaining singular values and the remaining eigenvalues of K
2
and KT by a positive gap. The same is true for Λ2 and K2 and KT .
Proof of Theorem 3.2.1. Because the entries K((t, σ), (t0 , σ0 )) are non-negative, the condition (3.14) with
r=1
means that
X
(t,σ
that is,
K
and
KT
K 2 ((t, σ), (t0 , σ 0 )) < ∞,
),(t0 ,
σ
(3.15)
0)
are Hilbert-Schmidt operators. It means that the operator
KKT
has an
orthonormal basis of eigenvectors and the series of squares of its eigenvalues (counted with
trT−C (KKT ). Consequently, the operators K
T
2
T 2
and K are bounded (and even completely bounded) and K and (K ) are of trace class.
2
The latter fact means that the matrix trace of the operator K coincides with its operator
2
T 2
2
trace in `T−C , and the same is true of (K ) . In addition, the operator K has the property
(2)
that its matrix entries K
((t, σ), (t0 , σ 0 )) are strictly positive:
multiplicities) converges and gives the trace
K (2) ((t, σ), (t0 , σ 0 )) =
X
e ))K((et, σ
e ), (t0 , σ 0 )) > 0.
K((t, σ), (et, σ
(3.16)
f)
(e
t,σ
0
The KreinRutman theory (see [KR48], Proposition VII or Appendix B) guarantees that
both
K and KT
have a maximal eigenvalue
Λ that is positive and non-degenerate, or simple.
3.2
THE TRANSFER-MATRIX
φ∗
K
AND ITS POWERS
KN
19
KT corresponding with Λ are
∗
unique up to multiplication by a constant, and all entries φ(t, σ) and φ (t, σ) are non∗
zero and have the same sign. In other words, the entries φ(t, σ) and φ (t, σ) can be made
positive. The spectral gaps are also consequences of the above properties. That is, the eigenvector
φ
of
K
and the eigenvector
Set:
s
λ(µ, β) = c2 (m2 + 1) (cosh 2β) 1 +
where
c
and
m
of
(m2 − 1)2
1
1−
(cosh 2β)2 (m2 + 1)2
!
(3.17)
are determined by
exp(β − µ)
− exp(β − µ))2 − e−2µ
m = e2β + (1 − e4β ) exp (−(β + µ)).
c =
Lemma 3.2.1.
(3.18)
e2β (1
(3.19)
For any β, µ > 0 such that
λ(µ, β) < 1,
the condition
(3.14)
(3.20)
is satised for r = 1:
tr(KKT ) = tr(KT K) < ∞ and
tr|K2 | = tr|(KT )2 | < ∞,
(3.21)
implying the assertions of Proposition 3.2.1 and Theorem 3.2.1. Moreover, the condition
(3.14) implies (3.20)
Proof of Lemma 3.2.1.
By denition the trace (3.21) we need to calculate the series
tr(KT K) =
X
KT K((t, σ), (t, σ))
(t,σ )
X
=
K((t, σ), (t0 , σ 0 ))K((t, σ), (t0 , σ 0 ))
(t,σ ),(t0 ,σ 0 )
X
=
(t,σ ),(t0 ,σ
A single-strip triangulation
t
K 2 ((t, σ), (t0 , σ 0 )).
consists of up- and down-triangles. Accordingly, it is con-
venient to employ new labels for spins: if a triangle
it by
tlup ;
σ(j)
l
by tdo ;
the corresponding spin
down-triangle then we denote it
the triangulation
t
(3.22)
0)
t(l)
is an l th up-triangle then we denote
will be denoted by
the spin
σ(j)
and
Similarly, if
t(j)
is an
lth
l
will be denoted by σdo . Consequently,
and its supported spin-conguration
t := (tup , tdo )
l
σup
.
σ
are represented as
σ := (σ up , σ do ).
20
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.2
TWO-DIMENSIONAL CDT
Here
tup = (t1up , . . . , tnup ), tdo = (t1do , . . . , tm
do ),
and
1
n
1
m
σ up = (σup
, . . . , σup
), σ do = (σdo
, . . . , σdo
),
assuming that the supporting single-strip triangulation
t
contains
n
up-triangles and
m
down-triangles. (The actual order of up- and down-triangles and supported spins does not
matter.)
(t0 , σ 0 ) as illustrated in
0
(t ∼ t ) i number of the
The same can be done for the pair
triangulations
t
and
t
0
that of up-triangles in
are consistent
(3.22). Let recall that the
down-triangles in
t
equals
t0 .
(t0 , σ 0 ) into a summation over
(t0up , σ 0up ) and (t0do , σ 0do ). Firstly, x a pair (t0up , σ 0up ) and make the sum over (t0do , σ 0do ). Note
0
0
0
that the term V ((t, σ), (t , σ )) depends only on σ do and σ up . Consequently,
To calculate the sum (3.22) we divide the summation over
X
t0do ,
σ
K 2 ((t, σ), (t0 , σ 0 ))
(3.23)
0
do
= e−βH(σ ) e−2βV ((t,σ ),(t ,σ )) e−µn(t)
0
0
X
e−βH(σ ) e−µn(t ) .
0
0
(t0do ,σ 0do )
The sum in the right-hand side of (3.23) can be represented in a matrix form. Denote by
e±1
the standard spin-1/2 unit vectors in
R2 :
 
 
e+1
Next, let us introduce a
2×2
 
1
 
= 
 
 
0
matrix

T
and
e−1
 
0
 
= .
 
 
1
where



 β



e−β 
e
t++ t+− 




T = e−µ 
 := 
.




 −β



β
e
e
t−+ t−−
Denote by
n(i), i = 1, . . . , nup (t0 )
the number of down-triangles in
(3.24)
t0
which are between the
3.2
ith
K
THE TRANSFER-MATRIX
and
(i + 1)th
t0 .
up-triangles in
Let
−βH(σ 0 ) −µn(t0 )
X
e
e
nup (t0 ) = k
=
=
n(i)≥0:
k Y
eT
M eσ0 l+1
σ 0 lup
up
−
l=1
where the matrix
M
k Y
P
i
KN
21
then
k Y
X
t0do ,σ 0do
AND ITS POWERS
eT
T n(l)+1 eσ0 l+1
σ 0 lup
up
n(i)≥1 l=1
eT
T eσ0 l+1
σ 0 lup
up
(3.25)
l=1
is the sum of the geometric progression




m++ m+− 


M=
T n := 
.


n=1


m−+ m−−
∞
X
(3.26)
Using the same procedure we can obtain the sum over all up-triangles into the triangulation
t. The only dierence is
nup (t0 ) = ndo (t) = k then
the existence of marked up-triangle in the strip: let as before
−βH(σ ) −µn(t)
X
e
e
=
tup ,σ up
k−1
Y
eT
l M eσ l+1
σup
up
2
1
eT
M
e
k
σup
σup
(3.27)
l=1
See Figure 3.1 for illustration of these calculations (3.25) and (3.27). Further, supposing the
existence of the matrix
M
and using (3.25) and (3.27) we obtain the following:
K 2 ((t, σ), (t0 , σ 0 )) = e−2βV ((tdo ,σ do ),(tup ,σ up ))
0
0
X
X
tup ,σ up t0do ,σ 0do
e−βH(σ ) e−µn(t)
X
×
tup ,σ up
X
e−βH(σ ) e−µn(t )
0
0
(t0do ,σ 0do )
= e−2βV ((tdo ,σ do ),(tup ,σ up ))
k hY
k−1
Y
T
T
T
2
1
M
e
×
eσ0 l M eσ0 l+1
e
e
M
e
l+1
l
k
σ
σup
σ
σ
up
up
0
0
up
l=1
k Y
−
do
eT
T eσ0 l+1
σ0 l
up
l=1
k−1
Y
up
l=1
do
eT
σ l M eσ l+1
do
do
i
2
1
M
e
.
eT
k
σup
σup
l=1
Necessary and sucient condition for the convergence of the matrix series for
the maximal eigenvalue of matrix
(3.28)
T
is less then 1. The eigenvalues of
λ± = e(β−µ) ± e−(β+µ) ,
T
M
is that
are
(3.29)
22
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.2
TWO-DIMENSIONAL CDT
mσ '1
up σ
(t ', σ ')
(t, σ )
2
'up
mσ '2 σ '3
up
2
σ 'up
σ '3up
σ 1do
2
σ do
3
σ do
do do
up
up
up
σ '1up
mσ 1 σ 2
mσ '3 σ '1
mσ 2 σ 3
do do
mσ(2)3 σ 1
do do
Figure 3.1:
Illustration of the calculates (3.25) and (3.27).
and the above condition means that
λ+ < 1
or, equivalently,
µ > ln 2cosh(β) .
Under this condition (3.30), the matrix
M=
M
(3.30)
is calculated explicitly:
e(β−µ)
e2β (1 − e(β−µ) )2 − e−2µ


(3.31)
 2β

1
e + (1 − e4β )e−(β+µ)



×
.




2β
4β −(β+µ)
1
e + (1 − e )e
We are now in a position to calculate the sum in (3.22). To this end, we again represent
it through the product of transfer matrices. Pictorially, we express the above sum as the
partition function of a one-dimensional Ising-type model where states are pairs of spins
l
l
(σdo
, σup
)
and the interaction is via the matrix
T
between the members of the pair and via
3.2
matrix
THE TRANSFER-MATRIX
M
K
AND ITS POWERS
between neighboring pairs. More precisely, dene the following
4×4

23
matrices:

 2β

m++ m+−
m+− m++
e2β m+− m+− 
e m++ m++








−2β
−2β
 m++ m−+

e
m
m
e
m
m
m
m
++
−−
+−
−+
+−
−−



Q=






−2β
−2β
e m−+ m+− e m−− m++
m−− m+− 
 m−+ m++






 2β

2β
e m−+ m−+
m−+ m−−
m−− m−+
e m−− m−−

(3.32)

 2β
(2)
(2)
(2)
(2) 
m++ m+−
m+− m++
e2β m+− m+− 
e m++ m++








(2)
(2)
(2)
(2)
−2β
−2β
 m++ m−+
e m++ m++ e m+− m−+
m+− m−− 



Qm = 




(2)
(2)
(2)
(2) 

e−2β m−+ m+− e−2β m−− m++
m++ m++ 
 m−+ m++






 2β
(2)
(2)
(2) 
(2)
2β
m−+ m−−
m−− m−+
e m−− m−−
e m−+ m−+


(3.33)


 2β
t++ m+−
t+− m++
e2β t+− m+− 
e t++ m++








−2β
−2β
 t++ m−+
e t++ m−− e t+− m−+
t+− m−− 



Qt = 






−2β
−2β
e t−+ m+− e t−− m++
t−− m+− 
 t−+ m++






 2β

e t−+ m−+
t−+ m−−
t−− m−+
e2β t−− m−−
Qtm
KN
(3.34)

 2β
(2)
(2)
(2)
(2) 
t++ m+−
t+− m++
e2β t+− m+− 
e t++ m++








(2)
(2)
(2)
(2)
−2β
−2β
 t++ m−+

e
t
m
e
t
m
t
m
++
+−
+−
++
−+
−−



=




(2)
(2)
(2)
(2) 

−2β
−2β
t
m
e
t
m
e
t
m
t
m
 −+ ++
−+ +−
−− ++
++ ++ 






 2β
(2)
(2)
(2)
(2) 
2β
e t−+ m−+
t−+ m−−
t−− m−+
e t−− m−−
(3.35)
24
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.2
TWO-DIMENSIONAL CDT
where
(2)
mij , mij
and
ti,j (i, j ∈ {−, +})
are elements of the matrices
M, M 2 ,
and
T
respec-
tively.
Now for the sum under consideration (3.22) we obtain using representation (5.53)
X
×
k Y
eT
M eσ0 l+1
σ 0 lup
up
l=1
k Y
−
k−1
Y
eT
T eσ0 l+1
σ0 l
up
= tr
eT
σ l M eσ l+1
do
do
l=1
k−1
Y
up
l=1
∞
X
0
0
(tdo ,σ do ),(t0up ,σ 0up )
(t,σ ),(t0 ,σ 0 )
"
e−2βV ((tdo ,σ do ),(tup ,σ up ))
X
K 2 ((t, σ), (t0 , σ 0 )) =
eT
σl
do
2
1
M
e
eT
k
σ
σup
up
2
1
M
e
M eσl+1 eT
k
σup
σup
#
do
l=1
k
Q
∞
X
Qm − tr
k=0
Qkt Qtm .
(3.36)
k=1
Qt elementwise. Thus the eigenvalue of
matrix Q is greater than the eigenvalue of the matrix Qt (it follows from the Perron-Frobenius
By the construction the matrix
Q
is greater then
theorem). Therefore the necessary and sucient condition for the convergence in (3.22) is
that the largest eigenvalue of
Q
is less than 1. It is possible to calculate its eigenvalue
Q
analytically. In order to express the eigenvalues of
and (3.19). In this notations the matrix
M,
it is convinient to use notations (3.18)
i.e. (3.31), is represented as following



 m 1

M = c


1 m



.


The equations for the eigenvalues of
Q
are:
λ1 = c2 eβ (m2 − 1)
λ2 = c2 e−β (m2 − 1)
s
λ3 = c2 (m2 + 1)(cosh β) 1 −
s
2
2
λ4 = c (m + 1)(cosh β) 1 +
(m2 − 1)2
1−
(cosh β)2 (m2 + 1)2
!
(m2 − 1)2
1−
(cosh β)2 (m2 + 1)2
!
A straightforward inspection conrms that the largest eigenvalue is given by
dition
λ4 < 1
coincides with (3.20). Finally, using matrices
(3.33)) with positive entries and of size
T
4 × 4,
tr(KK ) = tr
Q, Qm
λ4 .
The con-
(see formulas (3.32) and
we have the following representation of 3.22
X
k≥1
Q
k
Qm + . . . .
3.3
DISCUSSION AND OUTLOOK
The convergence of the matrix series
eigenvalue of the matrix
P
k≥1
25
Qk is equivalent to the condition that the maximal
Q is less then 1. This is exactly the condition (3.20). This completes
the proof of Lemma 3.2.1.
Theorem 3.2.2.
Under condition (3.20), the following limit holds:
1
log ΞN (β, µ) = log Λ.
N →∞ N
lim
(3.37)
Moreover, as N → ∞, the N -strip Gibbs measure PN (see Eqn (5.9)) converges weakly to
a limiting probability distribution P that is represented by a positive recurrent Markov chain
with states (t, σ), the transition matrix
P = {P ((t, σ), (t0 , σ 0 ))} and the invariant distribution π = {π(t, σ)} where
K((t, σ), (t0 , σ 0 ))φ(t0 , σ 0 )
Λφ(t, σ)
.
π(t, σ) = φ(t, σ)φT (t, σ) φ, φT T−C
P ((t, σ), (t0 , σ 0 )) =
2
P
with the norm φT−C = t,σ φ(t, σ)2 .
Proof of Theorem 3.2.2. The spectral gap for K implies that ∀ ψ ∈ `2T−C , we have the
convergence
1 N
K ψ = (hψ, φiT−C ) φ
N →∞ ΛN
lim
`2T−C . Moreover, let Π denote the operator of projection to the subspace
eigenvectors of K dierent from φ. Then
in the norm of space
spanned by the
1
kΠKPkT−C < 1 =⇒
Λ
N 1 = 0.
ΠKP
N
N →∞ Λ
T−C
lim
In turn, this implies that
1
1
log ΞN (µ, β) =
log trT−C KN → log Λ.
N
N
Convergence of the Gibbs measure
PN
follows as a corollary.
3.3 Discussion and outlook
This chapter makes a step towards determining the subcriticality domain for an Isingtype model coupled to two-dimensional causal dynamical triangulations (CDT). In doing
so we employ transfer-matrix techniques and in particular the Krein-Rutman theorem. We
complement the discussion of the previous sections with the following two concluding remarks:
26
TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO
3.3
TWO-DIMENSIONAL CDT
Remark 1.
It is instructive to summarise the logical structure of the argument establishing
Lemma 3.2.1 and Theorems 3.2.1 and 3.2.2:
• First,
(3.21)
holds i condition
(3.20)
holds: see the proof of Lemma 3.2.1.
• Next, (3.21) implies that K is a HilbertSchmidt operator and K2 is a trace class
operator in `2T−C .
• The last fact, together with the property of positivity (3.16), allow us to use the Krein
Rutman theory, deriving all assertions of Theorems 3.2.1 and 3.2.2.
On the other hand, if (3.20) fails (and therefore (3.21) fails), it does not necessarily mean
that the assertions Theorems 3.2.1 and 3.2.2 fail. In other words, we do not claim that the
boundary of the domain of parameters β and µ where the model exhibits uncritical behavior
is given by Eqn. (3.20). Moreover, Figure 3.2 shows the result of a numerical calculation
indicating that the condition (3.20) is worse than (3.11) for (moderately) large values of β .
An apparent condition closer to necessity is the pair of inequalities (3.14) for some (possibly) large r. This issue needs a further study.
Remark 2.
Physical considerations suggest that the critical curve in the (β, g) quarterplane would have some predictable patterns of behavior: as a function of β , it would decay
and exhibit a rst-order singularity at a unique point β = βcr ∈ (0, ∞).
A plausible conjecture is that the boundary of the critical domain coincides with the locus
of points (β, µ) where Λ looses either the property of positivity or the property of being a
simple eigenvalue. This direction also requires further research.
𝜆 ≤1
𝜆 ≤1
𝜇
𝛽≈0
𝛽
λQ = λ and λT are the maximal eigenvalues of the matrix Q and a related matrix T
respectively. The area above the black curve is where the condition (3.20) holds true.
Figure 3.2:
Chapter 4
FK representation for the Ising model
coupled to CDT
This chapter extends results from before chapter for the (annealed) classical Ising model
coupled to two-dimensional causal dynamical triangulations. Using the Fortuin-Kasteleyn
(FK) representation of quantum Ising models via path integrals, we determine a region in
the quadrant of parameters
β, µ > 0
where the critical curve for the classical model can be
located. In particular, we determine a region where the innite-volume Gibbs measure exists
and it is unique, and a region where the nite-volume Gibbs measure has no weak limit (in
fact, does not exist if the volume is large enough). We also provide lower and upper bounds
for the innite-volume free energy.
FK models were introduced by Fortuin and Kasteleyn (see [FK72]). These models have
become an important tool in the study of phase transition for the Ising and Potts model. The
goal of this chapter is to introduce the FK representation of a quantum Ising model coupled to
CDTs via a path integral (see [Aiz94], [Iof09] for an overview), and use this representation for
obtain information of the critical curve. The aforementioned FK representation uses a family
of Poisson point processes and the Lie-Trotter product formula to interpret exponential
sums of operators as random operator products. This representation was originally derived
in [Aiz94].
4.1 The quantum Ising model
In this section we write the classical partition function, over a given triangulation, by
using ingredients of the quantum Ising model.
Henceforth, for simplicity in notation and exposure of the following chapter, we shall
denote a triangle of any triangulation
t
doing without put the indices
i, j
as was done in
previous chapter. Thus, in this chapter, the Hamiltonian the (annealed) model is written as
follow
H(σ) = −
X
ht,t0 i
27
σ(t)σ(t0 ).
(4.1)
28
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
Here,
ht, t0 i stands that triangles t, t0
4.1
have a common edge. These triangles are called nearest
neighbor.
Let
t = (t(0), t(1), . . . , t(N − 1))
be a causal triangulation of
tial boundary condition (see Figure 3.1 (b)). Let
∆(t)
CN
with periodical spa-
denote the set of triangles of the
t.
We dene Ω(t) to be set of all spin congurations supported by the triangles of t, i.e.,
Ω(t) = {−1, +1}∆(t) . Let ZNβ,t be the partition function of the Ising model on the CDT t,
at inverse temperature β > 0
triangulation
X
ZNβ,t =
exp{−βH(σ)},
(4.2)
σ∈Ω(t)
where
H(σ)
σ ∈ Ω(t), dened by the formula
triangulation t is dened as follows.
represents the energy of conguration
The quantum Ising model on a causal
Let



 1 0

z
σ̂ = 


0 −1






(4.1).
(4.3)
be the Pauli matrix with their corresponding eigenvectors

φ+1
 
 1 
 
= 
 
 
0
In the quantum lenguage spins values
Notice that
z
σ̂ φν = νφν
for


±1
and
φ−1

 
 0 
 
=  .
 
 
1
(4.4)
are understood as eigenvalues of Pauli matrix.
ν = ±1.
t ∈ ∆(t) we associate a spin taking values φ+1 and φ−1 . Thus, the space
N
2
of all such spin congurations on t is dene as the real vector space Xt =
t∈∆(t) R , where
N
stands for the tensor product. Notice that Xt is a real vector space of dimension 2 to the
n(t) power: dim(Xt ) = 2n(t) .
For each classical conguration σ ∈ Ω(t) we associate the quantum conguration as
To each triangle
tensor products
φσ := ⊗t∈∆(t) φσ(t) ,
t ∈ ∆(t). Notice that there is a oneone correspondence between Ω(t) and the collection {φσ }σ∈Ω(t) . Moreover, the collection of
quantum congurations is a complete orthonormal basis of Xt with respect to the following
where
σ(t)
is the spin supported by the triangle
4.2
29
FK REPRESENTATION FOR ISING MODEL COUPLED TO CDT
scalar product
Y
hφσ |φσ0 i :=
φσ(t) , φσ0 (t)
2
,
t∈∆(t)
where
(·, ·)2
R2 .
is the usual scalar product of
z
linear self-adjoint operator σ̂ t
coordinate of
φσ
: Xt → Xt
With each triangle
t ∈ ∆(t)
we associate a
which acts as a copy of Pauli matrix
associated to the triangle
t
of
t.
That is, for each
σ̂ z
on the
σ ∈ Ωt ,
σ̂ zt φσ = φσ(t1 ) ⊗ · · · ⊗ σ̂ z φσ(t) ⊗ · · · = σ(t)φσ .
Note that operators
σ̂ zt , σ̂ zt0
(4.5)
commute, and satises
σ̂ zt σ̂ zt0 φσ = σ(t)σ(t0 )φσ .
The Hamiltonian
Ht
(4.6)
of the quantum Ising model is a linear self-adjoint operator dened on
Xt :
Ht = −
X
σ̂ zt σ̂ zt0 ,
(4.7)
ht,t0 i
where two operators
σ̂ zt
and
σ̂ zt0
interact if their supporting triangles
t, t0 ∈ ∆(t)
are nearest
neighbors.
Note that
Ht φσ = H(σ)φσ .
Ht
In other words,
is a diagonal in the
{φσ }
basis, and
corresponding eigenvalues being equal to values of the classical Ising Hamiltonian on cong-
σ . This allows write the classical partition function ZNβ,t
temperature β > 0 associated with triangulation t, as follows
urations
ZNβ,t =
X
X
exp{−βH(σ)} =
σ∈Ω(t)
for Ising model, at inverse
hφσ |e−βHt |φσ i = tr
e
−βHt
.
σ∈Ω(t)
Finally, using the quantum representation (4.8), the partition function for the
model coupled to CDT, at the inverse temperature
µ,
(4.8)
β>0
N -strip
Ising
and for the cosmological constant
can be written as follows
ΞN (β, µ) =
X
e
−µn(t)
tr e
−βHt
.
(4.9)
t
4.2 FK representation for Ising model coupled to CDT
In order to calculate tr e
−βHt
in (4.9) we will use the FK representation for the Ising
model via path integrals, see [Aiz94, Iof09]. By representation (4.9), the trace tr e
−βHt
may be expressed in terms of a type of path integral with respect to the continuous randomcluster model on
For any pair
∆(t)×[0, β] for any Lorentzian triangulation t (see Proposition 4.2.1 below).
t, t0 ∈ ∆(t)
of nearest neighbor triangles, we associate a Poisson process
30
4.2
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
ξht,t0 i (s)
on the time interval
of arrivals of operator
Kht,t0 i
[0, β]
with intensity
on the interval
Kht,t0 i =
2.
[0, β],
We refer to the process
ξht,t0 i
as process
where
I + σ̂ zt σ̂ zt0
.
2
(4.10)
ξ be the collection of independent Poisson processes ξht,t0 i : ξ(s) := {ξht,t0 i (s)}ht,t0 i∈Et ,
0
0
where Et is the set of all pairs of neighbor triangles: Et := {ht, t i : t, t ∈ ∆(t)}.
Let Pβ,t denote the probability measure associated with the family of Poisson process
ξ . We shall abuse notation by using ξ to denote a realization of process of arrivals ξ(s),
s ∈ [0, β]. By independence there are no simultaneous arrivals Pβ,t -a.s. Thus, a realization ξ
of process of arrivals can be represented by a collection of arrival times {si }i=1,...,Nξ contained
in [0, β] and its corresponding arrival types L(si ) ∈ Et , ξ ≡ {si , L(si )}i=1,...,Nξ , where Nξ is
the total number of arrivals during the time [0, β].
With a xed realization ξ we associated a family of all possible piecewise constant rightcontinuous functions ψ ξ = {ϕ : [0, β] → {φσ }}, having jumps only at arrival times of
ξ . Since Xt is nite dimensional and there are Pβ,t -a.s. nite number of arrivals, we have
|ψ ξ | < ∞, Pβ,t -a.s., where |ψ ξ | is the total number of functions in the set ψ ξ .
Let
s of a realization ξ corresponding a unique arrival type L(s) a.s.
0
Suppose that L(s) = ht, t i for t, t ∈ ∆(t) nearest neighbor, then KL(s) = Kht,t0 i : Xt → Xt .
−
Let ϕ ∈ ψ ξ , and denote ϕ(s ) = limt→s− ϕ(t). Notice that the function ϕ ∈ ψ ξ can be
continuous or not at each arrival time s (see Figure 4.1 below).
For each arrival time
0
arrival of
operator
arrival of
operator
A trajectory sample associated with a realization ξ = {si }i=1,...,n . Each trajectory
ϕ ∈ ψ ξ can be continuous or not at each arrival time s. In this case, at arrival time sk−1 the
trajectory ϕ do not have jump, and at arrival time sk the trajectory ϕ have a jump.
Figure 4.1:
Using the before notation, we have the following proposition.
4.2
31
FK REPRESENTATION FOR ISING MODEL COUPLED TO CDT
Proposition 4.2.1.
The matrix elements of the linear operator e−βHt with respect to the
basis {φσ } are given by
φσ |e−βHt |φσ0 = exp
3
βn(t)
2
Z
X
Pβ,t (dξ)
Y
hϕ(s− )|KL(s) |ϕ(s)i,
(4.11)
ϕ∈ψ ξ s∈ξ
ϕ(0)=φσ ,ϕ(β)=φσ0
for all t ∈ LTN .
Formula (4.11) was proved in [Aiz94] and [Iof09] for any general nite graph.
With any realization
ξ
we associate a graph
Gξ = (∆ξ , Eξ ),
where the set of vertices is
∆ξ = ∆(t) and the set of edges Eξ ⊆ Et is dened by following rule: an edge e = ht, t0 i ∈ Et
belong to Eξ if and only if there exist a arrival time s such that the corresponding arrival
0
type L(s) is ht, t i into the realization ξ .
0
0
We say that two triangulations t and t are connected, denoted by t ↔ t , if and only if
0
there exist a path in Gξ connecting t and t . For any t ∈ ∆(t), we suppose that t ↔ t. A subset
C ⊆ ∆(t) is called a cluster (maximal connected component) if for any t, t0 ∈ C then t ↔ t0 ,
0
0
and t = t for any t ∈ C and t ∈
/ C (see Figure 5.1 below). Thus, any realization ξ of the
Poisson process splits ∆(t) into the disjoint union of maximal connected components, i.e., for
any realization ξ there exists k = k(ξ) ∈ {1, . . . , n(t)} and sets C1 = C1 . . . , Ck = Ck(ξ) ⊆ ∆(t)
such that
k(ξ)
∆(t) =
[
Ci ,
i=1
and
Ci ∩ Cj = ∅
for
i 6= j ,
Additionally, we dene the
k(ξ) is the number of clusters dened by the relation ↔.
0
0
cluster Ct of a triangle t by Ct = {t ∈ ∆(t) : t ↔ t }.
Here
Time
In this gure, we show the Cluster Ct of a triangle t, and a graphic representation of
relation t ↔ t0 , where ↔ on right side in the gure, represent arrival times.
Figure 4.2:
Let
σ, σ 0 ∈ Ω(t)
be two congurations and let
φσ , φσ0
be the corresponding quantum
32
4.3
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
conguration. Then, for any
ht, t0 i ∈ Et
hφσ |Kht,t0 i |φσ0 i = δ{σ=σ0 } δ{σ(t)=σ(t0 )} .
Relation (4.12) implies that, for any realization
ξ
only constant functions of
the sum inside the integral (4.11). Additionally, an arrival of
imposes an additional condition
σ(t) = σ(t0 )
Ω(t, ξ) = {σ ∈ Ω(t) : σ has
Notice that
|Ω(t, ξ)| = 2k(ξ) ,
Kht,t0 i
ψξ
contribute to
at arrival time
s ∈ [0, β]
for contribute to the sum in (4.11). Dene
same sign in each cluster
Ci }.
and
Y
hϕ(s− )|KL(s) |ϕ(s)i =
X
(4.12)






1
















 0
s∈ξ
ϕ∈ ψ ξ
ϕ(0)=ϕ(β)=φσ
if
σ ∈ Ω(t, ξ)
(4.13)
if
σ∈
/ Ω(t, ξ)
As an elementary consequence of (4.13) the following representation for partition function
ZNβ,t
holds.
Proposition 4.2.2.
Let t ∈ LTN and β > 0. We have that
ZNβ,t
Proof.
= tr e
−βHt
= exp
Z
3
2k(ξ) Pβ,t (dξ).
βn(t)
2
(4.14)
The proof is consequence of Proposition 4.2.1 and equation (4.13).
Using the
N -strip Gibbs probability distribution PN,µ
(introduced in Eqn (2.9)) for pure
CDTs with periodical boundary condition, and substituting (4.14) on the right-hand side
of (4.9) we obtain the FK representation of partition function for the
coupled to CDTs, at inverse temperature
ΞN (β, µ) = ZN (r)
β>0
X Z
N -strip
Ising model
and for the cosmological constant
k(ξ)
2
µ
Pβ,t (dξ) PN,r (t),
t∈LTN
where
r = µ − 23 β
and
ZN (·)
is dened by (2.4).
4.3 The main results
This section contains the statement of the main theorems of the present chapter.
(4.15)
4.3
THE MAIN RESULTS
Understanding by
and
33
critical curve of the model the boundary of the domain of parameters β
µ where the model exhibits subcritical behavior, this chapter makes a rigorous derivation
of a subcriticality domain for an Ising model coupled to two-dimensional CDTs, and we nd
a domain where the tipical innite-volume Gibbs measure there no exists. In Figure 4.3, we
show a region where the critical curve of the model should be located. Formally, we dene
the critical curve as follow: We denote by
Gβ,µ
the set of
Gibbs measures
given by the closed
convex hull of the set of weak limits:
Pβ,µ = lim Pβ,µ
N ,
(4.16)
N →∞
and dene the domain of parameters where the weak limit Gibbs distribution exists
Γ = (β, µ) ∈ R2+ : Gβ,µ 6= ∅ ,
and domain where the weak limit Gibbs distribution exists and it is unique
Γ1 = (β, µ) ∈ R2+ : |Gβ,µ | = 1 .
It is evident that
Γ1 ⊆ Γ.
Thus, the critical curve
γcr
for the Ising model coupled to CDT is
dened by
γcr = ∂Γ1 ∩ R2+ .
Let
λ(β, µ)
(4.17)
be given by
s
λ(β, µ) = c2 (m2 + 1) (cosh 2β) 1 +
where
c
and
m
1
(m2 − 1)2
1−
(cosh 2β)2 (m2 + 1)2
!
(4.18)
are determined by
exp(β − µ)
− exp(β − µ))2 − e−2µ
m = e2β + (1 − e4β ) exp (−(β + µ)),
c =
e2β (1
(4.19)
(4.20)
Remember that identity (4.18) was derived in Chapter 3, Lemma 3.2.1.
We dene the strictly increasing function
ψ(β) = inf{µ ∈ R+ : λ(β, µ) < 1},
for
β > 0,
(4.21)
34
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
4.3
and the following set
S
3
2
Σ =
(β, µ) ∈ R+ : µ < − β + 2 ln 2
2
3
3
(β, µ) ∈ R2+ : µ < − β + ln
2
2
t1 , . . . , tk
Let
− 1 + ln 2 .
S × [0, 1] and σ 1 , . . . , σ k be their correspond0 ≤ i1 < · · · < ik ≤ N − 1 we dene the nite-dimensional
be triangulations of a single strip
ing spin congurations. Given
cylinder
e
2β
(t ,σ ),...,(tk ,σk )
1 1
Ci1 ,...,ik = Ci1 ,...,i
k
as follows
Ci1 ,...,ik = {(t, σ) : (t(i1 ), σ(i1 )) = (t1 , σ 1 ), . . . , (t(ik ), σ(ik )) = (tk , σ k )}
(4.22)
Theorem 4.3.1.
If (β, µ) ∈ Σ then there exists N0 ∈ N such that the partition function ΞN (β, µ) = +∞ whenever N > N0 . Moreover, the Gibbs distribution Pβ,µ
N with periodic
boundary conditions cannot be dened by using the standard formula with ΞN (β, µ) as a normalising denominator, consequently, there is no limiting probability measure Pβ,µ as N → ∞.
Formally, for any nite-dimensional cylinder Ci1 ,...,ik we obtain Pβ,µ
N (Ci1 ,...,ik ) = 0 whenever
N > N0 ≥ max{i1 , . . . , ik }.
Let
β1∗ , β2∗
be positive solution of equations
3
3
3
− β + 2 ln 2 = − β + ln(e2β − 1) + ln 2
2
2
2
and
3
β + 2 ln 2 = ψ(β),
2
(4.23)
(4.24)
respectively. Together with results from before chapter (see [HYSZ13] for more details),
Theorem 4.3.1 provides two-side bounds for the critical curve.
Theorem 4.3.2.
The critical curve γcr satises the following inequalities.
1. If (β, µ) ∈ γcr and 0 < β < β1∗ , then
3
− β + 2 ln 2 ≤ µ < ψ(β).
2
The above bound implies that: For any sequence {(βk , µk )} ⊂ γcr such that βk → 0,
then limk→∞ µk = 2 ln 2.
2. If (β, µ) ∈ γcr and β1∗ ≤ β < β2∗ , then
3
3
ln(e2β − 1) − β + ln 2 ≤ µ < ψ(β).
2
2
4.3
THE MAIN RESULTS
35
15
10
5
0
0
1
2
3
4
5
6
7
8
Figure 4.3: The area above the minimum of the dotted curve I (graph of the function ψ dened in
(4.21)) and dash-dotted line II is where the limiting Gibbs probability measure exists and is unique.
The critical curve lies in the region below the dotted curve I and dash-dotted line II but above the
continuous curve III and dashed line IV.
3. If (β, µ) ∈ γcr and β2∗ ≤ β < ∞, then
3
3
3
ln(e2β − 1) − β + ln 2 ≤ µ < β + 2 ln 2.
2
2
2
As a by-product of the proof of Theorems 4.3.1 and 4.3.2, using the FK representation
we also nd a lower and upper bound for the innite-volume free energy.
3
If µ > β + 2 ln 2, then the free energy for the innite-volume Ising model
2
coupled to CDTs is nite and satises the following inequalities.
Corollary 4.3.1.
1
ln 2, then
3
3
1
3
ln Λ µ + β − ln 2 ≤ lim
ln ΞN (β, µ) ≤ ln Λ µ − β − ln 2 .
N →∞ N
2
2
1. If 0 < β <
2. If
1
ln 2 ≤ β < ∞, then
3
1
3
3
ln Λ µ − β ≤ lim
ln ΞN (β, µ) ≤ ln Λ µ − β − ln 2 .
N →∞ N
2
2
Here Λ(s) is given by (2.10).
36
4.4
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
For each
N ∈ N,
we dene the follow set in
ΓN = {(β, µ) ∈ R2+ : KN
Γ− =
\
ΓN
R2+
is of trace class in
and
Γ+ =
N ∈N
Obviously,
N ≥ 1,
we
[
`2T−C },
(4.25)
ΓN .
(4.26)
N ∈N
Γ− ⊂ ΓN ⊂ Γ+ , for any N ≥ 1, and Pβ,µ
N there exist on ΓN . In order to each
dene the N -strip functions fN associated with the partition function for the N
-strip Ising model coupled to CDTs as
fN (β) = inf{µ ∈ R2+ : (β, µ) ∈ ΓN }
for
β > 0.
(4.27)
According to Theorem 3.2.2 in before chapter , Theorem 4.3.2 and Proposition 4.4.4,
given in the Section 4.4.2, implies a similar version of Theorem 3.2.2, as following.
Theorem 4.3.3.
For (β, µ) ∈ Γ+ = {(β, µ) ∈ R2+ : µ > fT−C (β)}, the following limit holds:
1
ln ΞN (β, µ) = ln Λ(β, µ),
N →∞ N
lim
(4.28)
where Λ(β, µ) is the maximal eigenvalue of K and KT in `2T−C and fT−C is pointwise limit
of the family of functions {fN }. Consequently, as N → ∞ the N -strip Gibbs measure Pβ,µ
N
converges weakly to a limiting probability distribution Pβ,µ .
4.4 Proof of Theorem 4.3.1 and 4.3.2
The proof is based on nding of upper and lower bounds for the functions
fN , introduced
in (4.27), using the FK representation (4.15) and the asymptotic behaviour of the partition
function
ZN (·)
for pure CDTs with periodical boundary condition. These bounds with the
Proposition 4.4.1, Proposition 4.4.2 and Proposition 4.4.3, established bounds for the critical
curve.
4.4.1
Proof of Theorem 4.3.1
We need two preparatory results. Let
1 ≤ i ≤ n(t),
t
be a Lorentzian CDT on cylinder
CN .
Given
we dene the sets
Πi = {all
realization
ξ
of process
{ξht,t0 i } such
that
k(ξ) = i}.
(4.29)
4.4
PROOF OF THEOREM
??
AND
??
37
Thus, we have the following representation of (4.14)
ZNβ,t
=
3
βn(t)
e2
n(t)
X
2i Pβ,t (Πi ).
(4.30)
i=1
ξ ∈ Πk and let {Cl }kl=1 be the corresponding cluster decomposition of the set ∆(t).
Let ηl = η(Cl ) and κl = κ(Cl ) denote the number of vertices (triangles) in cluster Cl and the
number of edges in Cl , respectively. Note that κl depends on the geometry of cluster Cl .
0
0
The probability that two nearest neighbor triangles t, t are linked is Pβ,t (t ↔ t ) =
1 − e−2β . Then, denoting p := 1 − e−2β , we obtain the following representation for the
probability of the set Πk ,
Let
Pβ,t (Πk ) =
X
p
P
l
κl
Pk
3
(1 − p) 2 n(t)−
l=1
κl
C1 ,...,Ck ⊆∆(t)
(4.31)
3
X
= (1 − p) 2 n(t)
C1 ,...,Ck ⊆∆(t)
p
1−p
Pkl=1 κl
.
Combining (4.31) with (4.30), we get the representation by cluster of the partition function
of Ising model supported by the triangulation
ZNβ,t
=e
− 23 βn(t)
n(t)
X
k=1
t
X
k
2
e
2β
Pkl=1 κl
−1
.
(4.32)
C1 ,...,Ck
In order to obtain lower bounds for the critical curve, we employ the representation (4.32)
and consider several particular cases of interest.
The case k = n(t) : In this case there exists an unique way to decompose the set ∆(t)
in n(t) maximal connected components, considering clusters as isolated vertices Cl = {t},
t ∈ ∆(t), and 1 ≤ l ≤ n(t). This decomposition implies that κl = κ(Cl ) = 0. Thus, by
relation (4.32), we obtain the following lower bounds for the partition function of the Ising
model on triangulation
t
3
ZNβ,t ≥ e(− 2 β+ln 2)n(t) .
Using (4.15), the lower bound in (4.33) provides the following lower bound to
3
ΞN (β, µ) ≥ ZN µ + β − ln 2 .
2
(4.33)
ΞN (β, µ),
(4.34)
Thus, using the asymptotic property given in Proposition (2.2.1) and Remark 2.2.2, we
38
4.4
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
obtain that the partition function
ΞN (β, µ)
there exists if
3
π
µ > − β + 2 ln 2 + ln cos
.
2
N +1
Letting
N →∞
we obtain the following proposition.
3
If (β, µ) ∈ R2+ such that µ < − β + 2 ln 2 then there exists N0 ∈ N
2
such that the partition function ΞN (β, µ) = +∞ whenever N > N0 . Moreover, the Gibbs
distribution Pβ,µ
N with periodic boundary conditions cannot be dened by using the standard
formula with ΞN (β, µ) as a normalising denominator, consequently, there is no limiting
probability measure Pβ,µ as N → ∞. Futhermore, for any nite-dimensional cylinder Ci1 ,...,ik
we obtain Pβ,µ
N (Ci1 ,...,ik ) = 0 whenever N > N0 ≥ max{i1 , . . . , ik }.
Proposition 4.4.1.
The case k = n(t) − 1 :
This case is discussed here for an illustrative purpose. Notice
that in this case there exists
connected components:
3
n(t) ways to decompose the set
2
n(t) − 1
in
n(t) − 1
maximal
isolated vertices (triangles) and one cluster of two nearest
neighbor vertices (triangles). That is, if
each decomposition
∆(t)
C1 , . . . , Cn(t)−1
C
η(C) = 1 or 2. Moreover, for
κl = 1 . This implies the following
is a cluster, then
we have that
Pn(t)−1
l=1
inequality
ZNβ,t >
X
1 (− 3 β+ln 2)n(t)
e 2
2
C ,...,C
1
e
2β
Pn(t)−1 κl
− 1 l=1
n(t)−1
=
3
4
e
2β
3
− 1 n(t)e(− 2 β+ln 2)n(t)
>
3
4
e
2β
3
− 1 e(− 2 β+ln 2)n(t) ,
as
(4.35)
n(t) ≥ 1.
Thus, we obtain another lower bound for the partition function of N -strip Ising model
coupled to CDTs
3
ΞN (β, µ) ≥
4
e
2β
− 1 ZN
3
µ + β − ln 2 .
2
(4.36)
Therefore, in this case we get the same inequality that in Proposition 4.4.1.
It would be interesting to analyse a general case
yield a better bound.
k = n(t) − l,
but it seems that it won't
4.4
PROOF OF THEOREM
The case k = 1 :
The probability of
=
(0)
Π1
number of edges in cluster
AND
??
39
Π1 :
Consider the following subset of
(0)
Π1
??
3
is κ1 = n(t) ∩ Π1 .
2
is easy to calculate
(0)
Pβ,t (Π1 ) = 1 − e−2β
23 n(t)
.
Then, by relatio (4.32)
3
ZNβ,t > 2e− 2 βn(t)
e
2β
3 n(t)
−1 2
3
3
= 2 exp −
β − ln
2
2
e
2β
− 1 n(t) .
Thus
ΞN (β, µ) > 2
P
te
−µn(t)
3
3
exp −
β − ln
2
2
e
2β
− 1 n(t)
(4.37)
3
3
= 2ZN µ + β − ln
2
2
e
2β
−1 .
As before, by asymptotic property (2.2.1), the partition function exists if
3
3
µ > − β + ln
2
2
Letting
N →∞
e
2β
− 1 + ln 2 cos
π
N +1
.
we obtain the following proposition.
3
3
If (β, µ) ∈ R2+ such that µ < − β + ln(e2β − 1) + ln 2 then there
2
2
exists N0 ∈ N such that the partition function ΞN (β, µ) = +∞ whenever N > N0 . Moreover,
the Gibbs distribution Pβ,µ
N with periodic boundary conditions cannot be dened by using the
standard formula with ΞN (β, µ) as a normalising denominator, consequently, there is no
limiting probability measure Pβ,µ as N → ∞. Futhermore, for any nite-dimensional cylinder
Ci1 ,...,ik we obtain Pβ,µ
N (Ci1 ,...,ik ) = 0 whenever N > N0 ≥ max{i1 , . . . , ik }.
Proposition 4.4.2.
Proof of Theorem 4.3.1.
sition 4.4.2.
The proof follows immediately from Proposition 4.4.1 and Propo-
40
4.4
FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT
4.4.2
Proof of Theorem 4.3.2
The proof of Theorem 4.3.2 relies on two aditional obsevations. These are:
(1) Upper bounds for the functions
fN
and existence of the pointwise limit
limN →∞ fN =
fT −C .
(2) The fact that graph of
fT −C
provides an upper bound for the critical curve.
Consequently, as by-product of Chapter 3 (see [HYSZ13]), we obtain the following assertions.
Proposition 4.4.3.
For all N ∈ N, the following property of functions fN is fullled:
1. If 0 < β < β2∗ , then
fN (β) ≤ ψ(β),
(4.38)
where β2∗ is positive solution of Eqn (4.24) and function ψ is introduced in Eqn (4.21).
2. If β2∗ ≤ β < ∞, then
Proposition 4.4.4.
3
fN (β) ≤ β + 2 ln 2.
2
(4.39)
Functions fN converge pointwise:
fT−C (β) := lim fN (β) for β > 0.
(4.40)
N →∞
Combining (4.38), (4.39) with Proposition 4.4.4 and letting
desired upper bound for the limit function






fT−C (β) ≤






fT −C
we obtain the
fT −C
ψ(β)








3


 fT−C (β) ≤
β + 2 ln 2
2
Since the graph of
N → ∞,
if
0 < β < β2∗
(4.41)
if
β2∗ ≤ β < ∞.
lies above the critical curve, the right-hand side of (4.41) provides
an upper bound for the critical curve.
Proof of Theorem 4.3.2.
The upper bound of Theorem 4.3.2 is consequence of Eqn (4.41).
The lower bound is consequence of Proposition 4.4.1 and 4.4.2. This concludes the proof of
Theorem 4.3.2.
Chapter 5
Potts model coupled to CDTs and FK
representation
In this chapter using a natural generalization of Ising model, we extend results from
before chapters for the (annealed) classical Ising model coupled to two-dimensional causal
Potts model. Whereas in Ising systems the spins on two dierent values, in the q -state Potts model q distinct values, represent
by the elements of the set {1, . . . , q}, are allowed on any vertex from the triangulation t.
dynamical triangulations. Such generalization is called of
In Chapter 3 and 4, the Ising model was dened putting spins on any triangles (faces), but
it is equivalent to put spins on any vertex of dual triangulation, dened in Section 5.2.2. Using
duality relation on a torus (periodic boundary condition), we provide a relation between the
free energy of Potts model coupled to CDTs and Potts model coupled to DUAL CDTs.
Additionally, using the high temperature expansion (and duality relation), we determined a
region where the critical curve can be located. This bound serves for Ising model case, and
improves the bounds found in the before chapters (Chapter 3: Theorem 4.3.1 and Theorem
4.3.2. Chapter 4: Lemma 3.2.1 and Theorem 3.2.2).
5.1 Introduction and main results of this chapter
A causal dynamical triangulation (CDT), introduced by Ambjørn and Loll (see [AL98]),
together with its predecessor a dynamical triangulation (DT), constitute attemps to provide
a meaning to formal expressions appearing in the path integral quantisation of gravity (see
[ADJ97], [AJ06] for an overview). The idea is to regularise the path integral by approximating the geometries emerging in the integration by CDTs. As a result, the path integral over
geometries is replaced with a sum over all possible triangulations where each conguration
is weighted by a Boltzmann factor
tion and
µ
e−µ|T | ,
with
|T |
standing for the size of the triangula-
being the cosmological constant. The evaluation of the partition function was
reduced to a purely combinatorial problem that can be solved with the help of the early
work of Tutte [Tut62, Tut63]; alternatively, more powerful techniques were proposed, based
41
42
5.1
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
on random matrix models (see, e.g., [FGZJ95]) and bijections to well-labelled trees (see
[Sch97, BDG02]).
From a physical point of view it is interesting to study various models of matter, such as
q -state Potts model, coupled to the CDT. The goal of this chapter is dene the q -state
Potts model coupled to CDTs and will use the FK representation for study this model. In this
the
case, the calculation of the partition function also reduces to a combinatorial problem. For
the
2-state
Potts model (Ising model) coupled to a CDT some progress has been recently
made on existence of Gibbs measures and phase transitions (see [AAL99], [BL07], [HYSZ13]
and [Her14] for details). In particular, using transfer matrix methods, the Krein-Rutman
theory and FK representation for the Ising model, [Her14] provides a region in the quadrant
of parameters
β, µ > 0
where the innite-volume free energy has a limit, providing results
on convergence and asymptotic properties of the partition function and the Gibbs measure.
Thus, FK-Potts models, introduced by Fortuin and Kasteleyn (see [FK72]), prove that these
models have become an important tool in the study of phase transition for the Ising and
q -state
Potts model.
In general, the FK-Potts model on a nite connected graph (not necessarily planar) is
a model of edges of the graphs, each edge is either closed or open. The probability of a
conguration is proportional to
p#open
where the edge-weight
model. For
q ≥ 1,
edges
p ∈ [0, 1]
(1 − p)#closed
edges
and the cluster-weight
q
#clusters
q ∈ (0, ∞)
,
are the parameters of the
this model can be extended to innite-volume where it exhibits a phase
pc (q), that depend on the geometry of the graph. In the
transition at some critical parameter
case of planar graphs, there is a connection between FK-Potts models on a graph and on its
dual with the same cluster-weight
q
and appropriately related edge-weight
p
p∗ = p∗ (p)
2
case of Z to
and
(Kramers-Wannier duality). For example, this relation leads in the particular
a natural conjecture: the critical point is the same as the so-called self-dual point satisfying
psd = p∗ (psd ),
proved by Beara and Duminil-Copin in [BC12].
In the case of a FK-Potts model dened on a causal dynamical triangulation
t
with pe-
riodic boundary condition, or equivalently dened on a torus (see Figure 3.1 for a geometric
representation), its dual, dened on
t∗ ,
is not a FK-Potts model; but will enough for our
purposes. This relation together with the Edwars-Sokal coupling, using
nd a relation between the parameters
parameters
(β ∗ , µ∗ )
(β, µ)
p = 1 − e−β , permits
of the Potts model coupled to CDT and the
of its dual for the innite-volume (thermodynamic limit).
In the present chapter, we prove the following duality relation.
Theorem 5.1.1.
Let q ≥ 2. The free energy of the q -state Potts model coupled to causal
5.1
INTRODUCTION AND MAIN RESULTS OF THIS CHAPTER
(a)
43
(b)
Illustrating the region where the critical curve for Potts model coupled CDTs and dual
CDTs can be located.
Figure 5.1:
dynamical triangulation and its dual satised the following duality relation
lim
N →∞
1
1
ln ΞN (β, µ) = lim
ln Ξ∗N (β ∗ , µ∗ )
N
→∞
N
N
(5.1)
where ΞN , Ξ∗N denote the partition function of the q -state Potts model coupled to CDT and
coupled dual CDT respectively (dened in Section 5.2.1), and
q
β = ln 1 + β
e −1
∗
,
µ∗ = µ −
Thus, (5.1) relates the free energy of the
free energy of the
q -state
q -state
3
ln(eβ − 1) + ln q.
2
(5.2)
Potts model coupled to CDTs and the
Potts model coupled to dual CDTs, and maps the high and low
temperature of the dual models onto each other.
We will use the duality relation of Theorem 5.1.1 and the high-temperature expansion
for the
q -state
Potts model for determine a region in the quadrant of parameters where the
critical curve for the
q -state
Potts model coupled CDTs and
q -state
Potts model coupled
dual CDTs can be located (see Figure 5.1).
Understanding by
β
and
µ (β
∗
and
µ
∗
critical curve
of the model the boundary of the domain of parameters
on its dual, respectively) where the model exhibits subcritical behavior
(see denition in Section 4.3 of Chapter 4), this chapter makes a rigorous derivation of the
subcriticality domain for an
q -Potts
model coupled to two-dimensional CDT and a domain
where the tipical innite-volume Gibbs measure there no exists. The proof involve two techniques: the duality relation (Theorem 5.1.1) and high-temperature expansion for the
q -state
Potts model. In Figure 5.1, we show the region where the critical curve of the model should
be located (gray region).
44
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
5.1
Dene the sets
√ 3
2
Σ =
(β, µ) ∈ R+ : µ < max ln(2 q), ln
2
e
β
− 1 + ln 2
,
and
Σ
∗
=
∗
∗
(β , µ ) ∈
R2+
3
: µ < max ln(2q), ln
2
∗
e
β∗
− 1 + ln 2
.
We prove the following long theorem for existence and no existence of Gibbs measure for
the model.
Theorem 5.1.2.
Let q ≥ 2.
1. Potts model coupled to CDTs. If (β, µ) ∈ Σ then there exists N0 ∈ N such that the partition function ΞN (β, µ) = +∞ whenever N > N0 . Moreover, the Gibbs distribution
Pβ,µ
N with periodic boundary conditions cannot be dened by using the standard formula with ΞN (β, µ) as a normalising denominator, consequently, there is no limiting
probability measure Pβ,µ as N → ∞. Furthermore, if (β, µ) satised
3
3
eβ − 1
β
2/3
µ > ln q + e − 1 + ln 2 − ln q + ln 1 + (q − 1)
,
2
2
q + eβ − 1
(5.3)
the innite-volume free energy exists, i.e. the following limit there exists:
1
ln ΞN (β, µ).
N →∞ N
lim
Moreover, as N → ∞, the Gibbs distribution Pβ,µ
N converges weakly to a limiting probβ,µ
ability distribution P .
2. Potts model coupled to dual CDTs. If (β ∗ , µ∗ ) ∈ Σ∗ then we have the same conclusion
∗ ∗
∗ ∗
for the the Gibbs distribution PβN ,µ , i.e. there is no limiting probability measure Pβ ,µ
as N → ∞. Furthermore, if (β ∗ , µ∗ ) satised
3
q 2/3 − 1
3 ∗
,
µ > β + ln 2 + ln 1 +
2
2
eβ ∗
∗
(5.4)
∗
∗
the innite-volume free energy exists and, as N → ∞, the Gibbs distribution PβN ,µ
∗ ∗
converges weakly to a limiting probability distribution Pβ ,µ .
As a byproduct, the Theorem 5.1.2 serves to nd lower and upper bounds for the innitevolume free energy. Moreover, in the case of
2-state
Potts model (Ising model), Theorem
5.1.2 extends earlier results from [Her14], [HYSZ13] and improves the approximation of the
curve in high temperature given in [AAL99]. In aditional, this approach allows to get a
better aproximation of the critical curve and check the asymptotic behavior of the critical
5.2
45
NOTATIONS
curve given in [AAL99], and it say that critical curve is asymptotic to
3
β
2
In Theorem 5.1.2, we nd a lower and upper curve that converges fast to
+ ln 2, for q ≥ 2.
3
β + ln 2.
2
5.2 Notations
In this section we rts introduce notations and give a summary of
q -state Potts model and
we dene the Potts model coupled to CDTs. Finally, we give a short review of the EdwardsSokal coupling. We refer to [MYZ01], [Gri06], [HYSZ13], for more details. We attempt at
establishing regions where the innite-volume free energy converges, yielding results on the
convergence and asymptotic properties of the partition function and the Gibbs measure.
5.2.1
A Potts model coupled to CDTs
t be a CDT on the cylinder CN with periodic boundary condition. Each triangulation
be view as a graph t = (V (t), E(t)) embedded on a torus. Potts spin systems are
Let
t
can
generalizations of the Ising model. Whereas in Ising systems the spins on two dierent
values, in the
{1, . . . , q},
q -state
q
Potts model
distinct values, represent by the elements of the set
are allowed on any vertex from the triangulation
sample space
V (t)
Ω(t) = {1, . . . , q}
model energy, where two spins
t.
We consider the product
q -state Potts
0
vertices t, t are
and we consider a usual (ferromagnetic)
σ(t)
and
σ(t0 )
interact if their supporting
connected by an common edge; such vertices are called nearest neighbors, and this property
is reected in the notation
on
t
ht, t0 i.
Thus, the Hamiltonian used for the
q -state
Potts model
is given by
h(σ) = −
X
δσ(t),σ(t0 ) .
(5.5)
ht,t0 i
The partition function for the
q -state
Potts model on
ZP (β, q, t) =
X
t
is dene by
n
o
exp −βh(σ) ,
(5.6)
σ
where the summation is over any congurations
measure on
t
σ ∈ {1, . . . , q}V (t) .
Thus, the
q -state
Potts
is dene as follows
µtβ,q (σ)
n
o
1
exp −βh(σ) .
=
ZP (β, q, t)
(5.7)
q -state Potts model on a xed t, we dene the partition
function for the q -state Potts model coupled to CDTs, at the inverse temperature β > 0 and
the cosmological constant µ, as follows
Using the partition function for the
ΞN (β, µ) =
X
t
n
o
exp −µn(t) ZP (β, q, t)
(5.8)
46
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
where
n(t)
the
N -strip
stands for the number of triangles in the triangulation
t.
5.2
Similarly, we introduce
Gibbs probability distribution associated with (5.8)
Pβ,µ
N (t, σ)
and we denote by
Gβ,µ
the set of
n
o
1
=
exp −µn(t) − βh(σ) .
ΞN (β, µ)
Gibbs measures
(5.9)
given by the closed convex hull of the set
of weak limits:
Pβ,µ = lim Pβ,µ
N ,
(5.10)
N →∞
q -state Potts model can be dened on a general lattice G. Therefore, it
∗
is possible dene the q -state Potts model sobre the dual t of the triangulation t (see next
∗
section for a formal denition of t ). The partition function for the q -state Potts model on
t∗ will denote by ZP (β ∗ , q, t∗ ). Finally, we dene the partition function for the q -state Potts
∗
∗
∗
model coupled to dual CDTs, ΞN (β , µ ) as follow
In general, the
Ξ∗N (β ∗ , µ∗ ) =
X
n
o
exp −µ∗ n(t) ZP (β ∗ , q, t∗ ).
(5.11)
t
5.2.2
The FK-Potts model on Lorentzian triangulations
We now turn to the FK representation of the
q -state
Potts model. The random cluster
model was originally introduced by Fortuin and Kasteleyn [FK72] and it can be understood
as an alternative representation of the
q -state
Potts model. This representation will be
referred to as the FK representation or FK-Potts model. We are interested in study FKPotts model on CDTs and dual CDTs, and nd a duality relation relation between the
parameters of the model on CDTs and its dual. In [HYSZ13], [Her14], the model was dened
putting spins on any triangle (faces), but it is equivalent to put spins on any vertex of dual
graph, in this case, dual triangulation. In this section we work with triangulations with
periodic boundary conditions, i.e., Lorentzian triangulations embedded in a torus
T
(see
G = (V, E) be a graph embedded in T, we obtain
we place a dual vertex within each face of G. For
Figure 3.1 (b)) and its dual. In general, let
G∗ = (V ∗ , E ∗ ) as follows:
∗
∗ ∗
each e ∈ E we place a dual e = hx , y i joining the two dual vertices lying in the two faces
∗
∗
of G abutting e. Thus, V is in one-one correspondence with the set of faces of G, and E is
∗
a one-one correspondence with E . For each Lorentzian triangulation t, we denote by t its
its dual graph
dual.
t = (V (t), E(t)) be a Lorentzian triangulation with periodic boundary condition,
where V (t), E(t) denote the set of vertices and edges, respectively. The state space for the
E(t)
0
FK-Potts model is the set Σ(t) = {0, 1}
, containing congurations that allocate 0 s and
10 s to the edge e = hi, ji ∈ E(t). For w ∈ Σ(t), we call an edge e open if w(e) = 1, and closed
if w(e) = 0. For w ∈ Σ(t), let η(w) = {e ∈ E(t) : w(e) = 1} denote the set of open edges.
Thus, each w ∈ Σ(t) splits V (t) into the disjoint union of maximal connected components,
which are called the open clusters of Σ(t). We denote by k(w) the number of connected
Let
5.2
47
NOTATIONS
Geometric representation of a dual Lorentzian triangulation t∗ with periodic spatial
boundary condition.
Figure 5.2:
components (open clusters) of the graph
of isolated vertices. Two sites of
(V (t), η(w)),
and note that
k(w)
includes a count
t are said to be connected if one can be reached from another
via a chain of open bonds.
The partition function of the FK-Potts model on
t with parameters p and q
and periodic
boundary condition is dened by

X
ZF K (p, q, t) =
Y

w∈Σ(t)
Thus, the FK-Potts measure on

t
(1 − p)1−w(e) pw(e)  q k(w) ,
e∈E(t)
is dene as follows

Φtp,q (w) =
(5.12)
1

ZF K (p, q, t)

Y
(1 − p)1−w(e) pw(e)  q k(w) .
(5.13)
e∈E(t)
We will use a similarly notation for the FK-Potts model on dual triangulation
ZF K (p∗ , q, t∗ ) and Φ
the partition
∗
parameters p and q , respectively.
denote by
with
t∗
p∗ ,q
5.2.3
t∗ .
We
function and the FK-Potts measure on
t∗
The relation between the Potts model and FK-Potts model:
Edwards-Sokal coupling
There are several ways to make the connection between the Potts and FK-Potts model.
The correspondence between the
q -state
Potts model and FK-Potts model was established
by Fortuin and Kasteleyn [FK72] (see also [ES88], [Gri06]). In a modern approach, these two
models are related via a coupling, i.e., coupled the two systems on a common probability
space. This coupling was introduced by Edwards-Sokal in [ES88].
Let
t
be a CDT on the cylinder
CN
with periodic boundary condition. We consider the
48
5.2
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Ω(t) × Σ(t) where Ω(t) = {1, 2, . . . , q}V (t)
measure Q on Ω(t) × Σ(t) is dene by
product sample space
Edwards-Sokal
Q(σ, w) ∝
Y
and
Σ(t) = {0, 1}E(t) .
The
(1 − p)δw(e),0 + pδw(e),1 δσi ,σj
e={i,j}∈E(t)
Theorem 5.2.1 (Edwards-Sokal [ES88]). Let q ∈ {2, 3, . . . }. Let p ∈ (0, 1) and t a CDT with
periodic boundary condition, and suppose that p = 1−e−β . If the conguration w is distributed
according to an FK-Potts measure with parameters (p, q) on t, then σ is distributed according
to a q -state Potts measure with inverse temperature β . Furthermore, the Edwards-Sokal
measure provides a coupling of µtβ,q and Φtp,q , i.e.
X
Q(σ, w) = µtβ,q (σ),
w∈Σ(t)
for all σ ∈ Ω(t), and
X
Q(σ, w) = Φtp,q (w),
σ∈Ω(t)
for all w ∈ Σ(t). Moreover, we have the relation between partition functions
ZF K (p, q, t) = e−β|E(t)| ZP (β, q, t).
5.2.4
(5.14)
Duality for FK-Potts model coupled to CDTs with periodic
boundary conditions
In this section we obtain a relation between the partition functions of FK-Potts model
on a triangulation
t
and its dual. This relation was studied by Beara and Duminil-Copin
for the FK-Potts model on
Z2
with free, wired and periodic boundary condition (see [BC12]
for details). We will view wich the dual of a FK-Potts model dened on a torus is a quasi
FK-Potts model, but it is not very dierent from one.
t and t∗ a CDT with periodic boundary condition and its dual. Each conguration
∗
w ∈ Σ(t) = {0, 1}E(t) gives rise to a dual conguration w∗ ∈ Σ(t∗ ) = {0, 1}E(t ) given by
w∗ (e∗ ) = 1 − w(e). That is, e∗ is declared open if and only if the corresponding bond e is
∗
closed. The new conguration w is called the dual conguration of w , and note that there
∗
exists an one-one correspondence between Σ(t) and Σ(t ). As in the Section 5.2.2, to each
∗
∗
∗
∗
∗ ∗
conguration w there corresponds the set η(w ) = {e ∈ E(t ) : w (e ) = 1} of its open
Let
edges.
Now, beginning of FK-Potts model on
triangulation
Let
o(w)
t,
we try to obtain the dual model on the dual
t∗ .
(resp.
c(w))
denote the number of open edges (resp. closed) of
w, k(w)
the
5.2
49
NOTATIONS
(a)
(b)
(c)
(a) Geometric representation of a net (b) Geometric representation of a cycle (c) None
of cluster of w is a net or a cycle
Figure 5.3:
number of connected components of
w,
and
f (w)
the number of faces delimited by
w,
i.e.
the number of connected components of the complement of the set of open bonds. We will
δ(w).
Call a connected component of w a net if it contains two non-contractible simple loops
γ1 , γ2 of dierent homotopy classes, and a cycle if it contain a non-contractible simple loops
γ1 non-contractible but is not a net (see Figure 5.3). These denitions were introduced in
[BC12]. In aditional, notice that every conguration w can be of one three types:
now dene an additional parameters
•
One of the cluster of
case, we let
•
w
is a net. Then no other cluster can be a net or a cycle. In that
δ(w) = 2;
One of the cluster of
w
is a cycle. Then no other cluster can be a net, but other cluster
can be cycles as well (in which case all the involved, simple loops are in the same
homotopy class) We then let
•
None of the cluster of
w
δ(w) = 1;
is a net or a cycle. We let
Using this denition for the parameter
δ,
δ(w) = 0.
we obtained the following version of Euler's
formula.
Proposition 5.2.1
.
(Euler's formula)
Let t a CDT with periodic boundary condition and
w ∈ {0, 1}E(t) . Then
|V (t)| − o(w) + f (w) = k(w) + 1 − δ(w).
(5.15)
Using duality and Proposition 5.2.1, we have the following relations
o(w) + o(w∗ ) = |E(t)|,
f (w) = k(w∗ )
and
δ(w) + δ(w∗ ) = 2.
(5.16)
50
5.2
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Let
q ∈ (0, ∞)
and
p ∈ (0, 1).
The partition function of the FK-Potts model is given by


X
ZF K (p, q, t) =
Y

w∈Σ(t)
X
=
(1 − p)1−w(e) pw(e)  q k(w)
e∈E(t)
po(w) (1 − p)c(w) q k(w) .
w∈Σ(t)
Using Euler's formula and relations (5.16), we rewrite the number of cluster of
of its dual
w
in terms
w∗
k(w) = |V (t)| − |E(t)| + o(w∗ ) + k(w∗ ) + 1 − δ(w∗ ).
We note also that
o(w) + o(w∗ ) = |E(t)| = |E(t∗ )|.
Plugging before relations into the
partition function of the FK-Potts model, we obtain
ZF K (p, q, t) =
X
po(w) (1 − p)|E(t)|−o(w) q k(w)
w∈Σ(t)
= (1 − p)
X p o(w)
q k(w)
1−p
|E(t)|
w∈Σ(t)
X p |E(t)|−o(w∗ )
∗
∗
∗
q |V (t)|−|E(t)|+o(w )+k(w )+1−δ(w )
1−p
|E(t)|
= (1 − p)
w∈Σ(t)
= p
|E(t)| |V (t)|−|E(t)|
q
X 1 − p o(w∗ )
∗
∗
∗
q o(w )+k(w )+1−δ(w )
p
w∈Σ(t)
As there exists an one-one correspondence between
we change the sum in
Σ(t)
and
Σ(t∗ ),
in the last equality, we
Σ(t) by the sum in Σ(t∗ ). Thus, we obtain the following representation
of the partition function in terms of the dual triangulation and dual congurations
ZF K (p, q, t) = p
|E(t)| |V (t)|−|E(t)|
q
X q(1 − p) o(w∗ )
∗
∗
q k(w )+1−δ(w )
p
∗
∗
(5.17)
w ∈Σ(t )
Using the relation (5.17), we obtain the following lemma.
Lemma 5.2.1.
Let t be a CDT with periodic boundary condition. Then the following comparison inequalities both
ZF K (p, q, t) ≤
p
1 − p∗
|E(t)|
q |V (t)|−|E(t)|+1 ZF K (p∗ , q, t∗ )
(5.18)
5.2
NOTATIONS
and
p
1 − p∗
|E(t)|
q |V (t)|−|E(t)|−1 ZF K (p∗ , q, t∗ ) ≤ ZF K (p, q, t)
51
(5.19)
where ZF K (p∗ , q, t∗ ) is the partition function for FK-Potts model on t∗ with parameters q
and p∗ = p∗ (p, q) satisfying
(1 − p)q
p∗
p
p (p, q) =
, or equivalently
= q.
·
∗
(1 − p)q + p
1−p 1−p
∗
Proof.
We introduce the parameter
p∗ = p∗ (p, q)
as solution of the equation
p∗
(1 − p)q
.
=
∗
1−p
p
Thus, the partition function can be written in the following ways
X p∗ o(w∗ )
∗
∗
q k(w )+1−δ(w )
∗
1−p
∗
∗
ZF K (p, q, t) = p|E(t)| q |V (t)|−|E(t)|
w ∈Σ(t )
p|E(t)|
∗
|V (t)|−|E(t)|
=
(1 − p∗ )|E(t )|
∗ )| q
∗
|E(t
(1 − p )
Notice that
−1 ≤ 1 − δ(w∗ ) ≤ 1,
X p∗ o(w∗ )
∗
∗
q k(w )+1−δ(w ) .
∗
1−p
∗
∗
w ∈Σ(t )
w∗ ∈ Σ(t∗ ). We dene ZF K (p∗ , q, t∗ ), the partition
∗
parameters p and q . Thus, we obtain the upper bound
for all
function of a FK-Potts model with
p|E(t)|
|V (t)|−|E(t)|+1
ZF K (p∗ , q, t∗ ) ,
ZF K (p, q, t) ≤
∗ )| q
∗
|E(t
(1 − p )
and the lower bound
p|E(t)|
q |V (t)|−|E(t)|−1 ZF K (p∗ , q, t∗ ) ≤ ZF K (p, q, t)
(1 − p∗ )|E(t∗ )|
for the partition function of FK-Potts model on
correspondence between
E(t)
and
E(t∗ ),
t with parameters p and q . Using the one-one
we conclude the proof.
The partition function for pure CDT's has been determined as a sum over all possible
triangulations of a cylinder where each conguration is weighted by a Boltzmann factor
e−µn(t) ,
where
n(t)
standing for the size of the triangulation and
constant. Thus, in quantum gravity the volume
n(t)
µ
being the cosmological
becomes a dynamical variable for the
model. Therefore, we rewrite the duality relation ( Lemma 5.2.1) for the partition function
t, in terms of dynamical variable n(t). In the Table
∗
∗
5.1 we show the relation among V (t), E(t), V (t ), E(t ) and the number of triangles n(t) of
a CDT t.
of the FK-Potts model on a triangulation
52
5.3
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
t = (V (t), E(t))
|V (t)| =
1
n(t)
2
|E(t)| =
3
n(t)
2
|faces
in
Table 5.1:
t| = n(t)
t∗ = (V (t∗ ), E(t∗ ))
|V (t∗ )| = n(t)
|E(t∗ )| =
|faces
in
3
n(t)
2
t∗ | =
1
n(t)
2
Relation between the graphs t, t∗ and n(t)
Using the relations of Table 5.1, the Lemma 5.2.1 becomes be written in terms of
n(t)
as follow
Corollary 5.2.1.
Let t be a CDT with periodic boundary condition. Then the following
comparison inequalities both
and
p
1 − p∗
p∗
1−p
23 n(t)
q
32 n(t)
q
−1−n(t)
−1− 12 n(t)
ZF K (p, q, t)
≤
≤
ZF K (p∗ , q, t∗ )
ZF K (p∗ , q, t∗ )
≤
≤
ZF K (p, q, t)
p
1 − p∗
p∗
1−p
32 n(t)
32 n(t)
q 1−n(t)
(5.20)
1
q 1− 2 n(t)
(5.21)
where ZF K (p∗ , q, t∗ ) is the partition function for FK-Potts model on t∗ with parameters q
and p∗ = p∗ (p, q) satisfying
p∗ (p, q) =
(1 − p)q
p∗
p
, or equivalently
= q.
∗
(1 − p)q + p
1−p 1−p
5.3 The proof of Theorem 5.1.1 and rst bounds for the
critical curve
In the previous section we found comparison inequalities between the partition function
of the FK-Potts model on
t and the partition function of the FK-Potts model on its dual t∗ .
In this section we will use these comparison inequalities to prove Theorem 5.1.1. Combining
inequalities (5.20), (5.21) and the Edwars Sokal coupling (Theorem 5.2.1), we obtain the
following comparison inequalities between the partition function of the
on
t
and the partition function of the
p
1 − p∗
32 n(t)
q
−1−n(t)
e
3
(β−β ∗ )n(t)
2
q -state
Potts model on its dual
ZP (β, q, t)
≤
≤
ZP (β ∗ , q, t∗ )
p
1 − p∗
32 n(t)
q -state
Potts model
t∗ .
3
q 1−n(t) e 2 (β−β
∗ )n(t)
(5.22)
5.3
53
THE PROOF OF THEOREM 5.1.1 AND FIRST BOUNDS FOR THE CRITICAL CURVE
and
p∗
1−p
32 n(t)
q
−1− 12 n(t)
e
3
(β ∗ −β)n(t)
2
ZP (β ∗ , q, t∗ )
≤
≤
ZP (β, q, t)
p∗
1−p
32 n(t)
1
3
q 1− 2 n(t) e 2 (β
∗ −β)n(t)
(5.23)
where
β∗
(eβ − 1)(e
− 1) = q .
Proof of Theorem 5.1.1.
Using the comparison inequalities (5.22) and (5.23), we will nd
comparison inequalities for the partition functions of the Potts model coupled CDTs with
β, µ and dual CDTs with parameters β ∗ = β ∗ (β), µ∗ = µ∗ (β, µ). Remember that
∗
p∗ = 1 − e−β and p = 1 − e−β . Thus,
parameters
q
p∗
∗
∗
= (1 − e−β ) + qe−β =
−β
1−p
(1 − e ) + qe−β
and
p
q
=
= (1 − e−β ) + qe−β .
∗
∗
−β
1−p
(1 − e ) + qe−β ∗
Multiplying by the Boltzmann factor
CDTs of the cylinder
CN ,
e−µn(t)
in (5.22) and (5.23), and sum over all possible
we obtain the following comparison inequalities
1 ∗ ∗ ∗
Ξ (β , µ ) ≤ ΞN (β, µ) ≤ qΞ∗N (β ∗ , µ∗ )
q N
where
Ξ∗N
stands the partition function of the
with periodic boundary condition,
ΞN
q -state
(5.24)
Potts model coupled to dual CDTs
stands the partition function of the
q -state
Potts
model coupled to CDTs with periodic boundary condition, and
β = ln 1 +
∗
Similarly, we have
where
q
β
e −1
,
µ∗ = µ −
3
ln(eβ − 1) + ln q.
2
1
ΞN (β, µ) ≤ Ξ∗N (β ∗ , µ∗ ) ≤ qΞN (β, µ)
q
q
β = ln 1 + β ∗
e −1
,
µ = µ∗ −
(5.25)
3
1
∗
ln(eβ − 1) + ln q.
2
2
Take the natural logarithm in inequalities (5.24) and (5.25), divide both sides of the above
inequalities by
N
and let
N → ∞.
This concludes the proof of Theorem 5.1.1.
q -state
of the q -
Theorem 5.1.1 provide an interesting reformulation in terms of free energy of the
Potts model coupled to CDTs and its dual. This theorem relates the free energy
state Potts model coupled CDTs and the free energy of the
q -state Potts model coupled dual
CDTs, and maps the high and low temperature of the dual models onto each other.
54
5.3
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Using Edwars-Sokal coupling for the partition functions (5.14), duality relation found in
Theorem 5.1.1 and asymptotic properties (2.14), (3.20) of the partition function for pure
CDTs (see [MYZ01] for more details), we will obtain the rst bounds for the critical curve
of the
q -state
t be a CDT with periodic
Σ(t) which splits V (t) in i
Potts model coupled to CDTs and dual CDTs. Let
boundary condition. We dene the set
Πi
of congurations in
maximal connected components, i.e.
Πi = {w ∈ Σ(t) : k(w) = i}.
Similarly, we denote
Π∗i
Σ(t∗ )
the set of congurations in
which splits
V (t∗ )
in
i
maximal
connected components. Thus, we have the following representation for the partition function
of
q -state
Potts model on
t
|V (t)|
ZP (β, q, t) =
eβ|E(t)| φtp (q k(w) )
β|E(t)|
=e
X
q i φtp (Πi ),
(5.26)
i=1
and on
t∗
|V (t∗ )|
∗
∗
ZP (β , q, t ) =
∗
∗
∗
eβ|E(t )| φtp (q k(w ) )
β|E(t∗ )|
=e
X
∗
q i φtp∗ (Π∗i ),
(5.27)
i=1
where
t∗
p∗
φtp , φ
denotes product measures on
Σ(t)
and
Σ(t∗ ),
respectively.
Using Table 5.1, we write the representations (5.26), (5.27) for the partition function in
terms of the dynamical variable
n(t). For that, we consider two cases of interest separately.
The model on CDTs t: In this case, we
volume n(t) of the triangulation as follow
1.
can to write the partition function in terms of
1
n(t)
2
ZP (β, q, t) = e
3
βn(t)
2
X
q i φtp (Πi ).
(5.28)
i=1
Using the rst and latter term on the right-hand side of (5.28), we obtain two lower bounds
for the partition function of the Potts model on
t
3 n(t)
ZP (β, q, t) ≥ q eβ − 1 2
,
These lower bounds for the
q -state
Potts model on
1
ZP (β, q, t) ≥ q 2 n(t) .
t
(5.29)
permit to obtain a lower barrier for
parameters where the model can be dened, and the partition function of the model coupled
to CDTs could no explode in nite volume. These lower bounds serves to obtain information
of the Gibbs measure for
q -state
Potts model coupled to CDTs.
5.3
THE PROOF OF THEOREM 5.1.1 AND FIRST BOUNDS FOR THE CRITICAL CURVE
Proposition 5.3.1.
55
If (β, µ) ∈ R+ such that
µ<
3
1
ln q + ln 2 or µ < ln(eβ − 1) + ln 2,
2
2
then there exists N0 ∈ N such that the partition function ΞN (β, µ) = ∞ whenever N >
N0 . Moreover, the Gibbs distribution Pβ,µ
N cannot be dened by using the standard formula
with ΞN (β, µ) as a normalising denominator, consequently, there is no limiting probability
measure Pβ,µ as N → ∞.
Proof. The
ΞN (β, µ),
ZP (β, q, t)
lower bounds in (5.29) to
ΞN (β, µ) ≥ q
X
e−{µ− 2 ln(e
3
)}n(t) ,
β −1
provide the following lower bounds to
ΞN (β, µ) ≥ q
t
X
e−{µ− 2 ln q}n(t) .
1
(5.30)
t
Using asymptotic properties of Proposition (2.2.1), we obtain which the partition function
ΞN (β, µ)
there is no exist if
π
1
µ ≤ ln q + ln 2 cos
2
N +1
Letting
N → ∞,
π
3
β
ln(e − 1) + ln 2 cos
or µ ≤
.
2
N +1
we conclude the proof.
Now, notice that
1
n(t)
2
X
1
q i φtp (Πi ) ≤ q 2 n(t) ,
i=1
t. Thus,
on t,
for any triangulation
q -state
Potts model
we obtain a upper bound for the partition function of the
3
1
3
1
ZP (β, q, t) ≤ e 2 βn(t) q 2 n(t) = e( 2 β+ 2 ln q)n(t) .
q -state Potts model on t, permit to obtain a rst
the q -state Potts model coupled to CDTs, and above
(5.31)
This upper bound for the
upper barrier
for the critical curve of
of that upper
bound the model exhibits subcritical behavior. Moreover, this upper bound for the critical
curve of the model coupled serves to obtain information of the Gibbs measure for the
q -state
Potts model coupled to CDTs. We get the following result.
3
1
Under condition µ > β + ln q + ln 2, the innite-volume free energy
2
2
exists, i.e. the following limit there exists:
Proposition 5.3.2.
1
ln ΞN (β, µ).
N →∞ N
lim
Moreover, as N → ∞, the Gibbs distribution Pβ,µ
N converges weakly to a limiting probability
distribution Pβ,µ .
56
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Proof.
Using inequality (5.31), we get to
ΞN (β, µ) ≤
X
5.3
e−{µ− 2 β− 2 ln q}n(t)
3
1
t
By Proposition 2.2.1, we obtain which the free energy there exists if
µ − 32 β − 21 ln q > ln 2.
This concludes the proof.
Finally, using the duality relation (Theorem 5.1.1) and bounds found before propositions,
q -state
we obtain bounds for the critical curve for the
from CDTs
Table 5.2:
on its dual.
by duality
−−−−−−−−→
t
1
ln q + ln 2
2
→
µ<
3
ln(eβ − 1) + ln 2
2
→
µ>
1
3
β + ln q + ln 2
2
2
→
µ<
Potts model coupled to dual CDTs.
to dual CDTs
µ∗ <
t∗
∗
3
ln(eβ − 1) + ln 2
2
µ∗ < ln q + ln 2
∗
3
ln(q + eβ − 1) + ln 2
2
µ∗ >
Bounds for the critical curve of the q -state Potts model on CDTs will generate bounds
In Table 5.2 the parameters
(β, µ)
and
(β ∗ , µ∗ )
The model on dual CDTs t∗ : Similarly,
volume n(t) of the triangulation as follow
2.
∗
∗
ZP (β , q, t ) = e
satised the duality relation (5.2).
we write the partition function in terms of
3 ∗
β n(t)
2
n(t)
X
∗
q i φtp∗ (Π∗i ).
(5.32)
i=1
Using the rst and latter term on the right-hand side of (5.32), we obtain two lower bounds
for the partition function of the
q -state
Potts model on
3 n(t)
∗
ZP (β ∗ , q, t∗ ) ≥ q eβ − 1 2
,
t∗
ZP (β ∗ , q, t∗ ) ≥ q n(t) ,
(5.33)
and an upper bound
3
ZP (β ∗ , q, t∗ ) ≤ e 2 β
∗ n(t)
3 ∗
q n(t) = e( 2 β +ln q)n(t) .
(5.34)
Using asymptotic properties of Theorem 2.2.1 and Proposition 2.2.1, bounds (5.33) and
(5.34) provide bounds for the critical curve of the
q -state
Potts model coupled dual CDTs.
As in before case, we have the following proposition to existence and non existence of Gibbs
measures of the model.
5.4
THE PROOF OF THEOREM 5.1.1 AND FIRST BOUNDS FOR THE CRITICAL CURVE
Proposition 5.3.3.
57
For q -state Potts model coupled to dual CDTs, we have the following
assertions:
1. If (β ∗ , µ∗ ) ∈ R+ such that
µ∗ < ln q + ln 2 or µ∗ <
3
∗
ln(eβ − 1) + ln 2,
2
then there exists N0 ∈ N such that the partition function Ξ∗N (β ∗ , µ∗ ) = ∞ whenever
∗ ∗
N > N0 . Moreover, the Gibbs distribution PβN ,µ cannot be dened by using the standard formula with Ξ∗N (β ∗ , µ∗ ) as a normalising denominator, consequently, there is no
∗ ∗
limiting probability measure Pβ ,µ as N → ∞.
3
2. Under condition µ∗ > β ∗ + ln q + ln 2, the innite-volume free energy exists, i.e. the
2
following limit there exists:
1
lim
ln Ξ∗N (β ∗ , µ∗ ).
N →∞ N
∗
∗
Moreover, as N → ∞, the Gibbs distribution PβN ,µ converges weakly to a limiting
∗ ∗
probability distribution Pβ ,µ .
Finally, using the duality relation (Theorem 5.1.1), and bounds found in the before proposition, we obtain bounds for the critical curve for the Potts model coupled to CDTs. In Table
from dual CDTs
µ∗ <
t∗
by duality
−−−−−−−−→
µ∗ < ln q + ln 2
→
3 β∗
ln e − 1 + ln 2
2
→
3 ∗
β + ln q + ln 2
2
→
µ∗ >
to CDTs
µ<
3
ln eβ − 1 + ln 2
2
µ<
µ>
t
1
ln q + ln 2
2
3
ln q + eβ − 1 + ln 2
2
Bounds for the critical curve of the q -state Potts model coupled to dual CDTs will
generate bounds for the the critical curve of the q -state Potts model coupled to CDTs.
Table 5.3:
5.3, parameters
(β, µ)
and
(β ∗ , µ∗ )
satisfy the duality relation (5.2).
Tables 5.2 and 5.3 show that to nd bounds for the critical curve for the model on CDTs
provide bounds for the model on dual CDTs, and viceversa. Thus, in the next section we
improves the bounds obtained for the critical curve of the Potts model on CDTs. In aditional,
this approach allows to get a asymptotic behavior of the critical curve for the model on CDTs
and its dual.
58
5.4
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
5.4 High-T expansion of the Potts model and Proof of
Theorem 5.1.2
Let
t
model on
be a CDT with periodic boundary condition. The partition function for the Potts
t
is write in the usual high-T expansion as
ZP (β, q, t) =
q+h
q
|E(t)| X Y
σ
(1 + fij )
(5.35)
hi,ji
P
h
h = eβ − 1 and fij = q+h
(−1 + qδσi ,σj ). It can be readily veried that σ fij = 0 for
{i, j} ∈ E(t), consequently, all subgraphs with one or more vertices of degree 1 give rise
where
all
to zero contributions. Thus, the partition function can be written as follow
ZP (β, q, t) =
where
q+h
q
|E(t)| X X
σ
Y
fij ,
A∈G(t) {i,j}∈A
G(t) is the set of families of edges of t without vertices of degree 1. Therefore, we can
rewrite the partition function as
ZP (β, q, t) =
where
w(A) =
X Y
σ
w(A)
|E(t)| X
w(A)
A∈G(t)
is a weight factor associated with the subset
A.
We then pro-
{i,j}∈A
ceeded to determine
expanding in
fij
q+h
q
w(A).
An expression of
the product
X Y
σ
w(A)
fij .
for general
A
can be obtained by further
This procedure leads to
{i,j}∈A
w(A) =
h
q+h
|A| X
P(A)(σ),
σ
Q
P(A)(σ) = e∈A (−1 + qδe (σ)), and if e = {i, j}
P(A)(σ), we have the following representation
where
P(A)(σ) = (−1)|A| + (−1)|A|−1 q
X
X
e1 ,...,e|A|−1 ∈A
+q |A| δe1 (σ) . . . δe|A| (σ).
δe (σ) = δσi ,σj .
δe (σ) + (−1)|A|−2 q 2
e∈A
+ · · · + (−1)q |A|−1
then
X
e1 ,e2 ∈A
δe1 (σ) . . . δe|A|−1 (σ)
Expanding
δe1 (σ)δe2 (σ)
5.4
HIGH-T EXPANSION OF THE POTTS MODEL AND PROOF OF THEOREM 5.1.2
59
Examples of three subgraphs of A with 8 edges. It is clear that the term ξ(e1 , . . . , e8 )
depends of the topology of the subgraphs.
Figure 5.4:
We choose
k
edges
{e1 , . . . , ek }
X
of
A.
These edges form a subgraph of
A.
Thus, we obtain
δe1 (σ) . . . δek (σ) = q |V (t)|−k+ξ(e1 ,...,ek )
σ
ξ(e1 , . . . , ek ) stands the total numbers of internal faces in each maximal connected
component of {e1 , . . . , ek } (number of independent circuits in {e1 , . . . , ek }). Note that this
terms depends essentially on the topology of {e1 , . . . , ek } (see Figure 5.4). But ξ(e1 , . . . , ek ) ≤
2
(k + 1) for all k . Thus, we obtain the estimate
3
X
2
k
2
δe1 (σ) . . . δek (σ) ≤ q |V (t)|−k+ 3 (k+1) = q |V (t)|− 3 + 3
where
σ
and
X
σ
|A| |V (t)|− k + 2
3
3.
δe1 (σ) . . . δek (σ) ≤
q
k
∈A
X
e1 ,...ek
60
5.4
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Therefore,
X
P(A)(σ) ≤ q
|V (t)|+ 23
σ
|A| X
|A|
k
k=0
p
2 p
(−1)|A|−k ( 3 q 2 )k = q |V (t)|+ 3 ( 3 q 2 − 1)|A| ,
and
ZP (β, q, t) ≤
≤
where
q+h
q
|E(t)|
q+h
q
|E(t)|
q
|V (t)|+ 32
2
q |V (t)|+ 3
|A|
X p
h
3
2
( q − 1)
q+h
A∈G(t)
!
X
1+
Ωk (t)uk ,
k≥1
Ωk (t) = |{A ∈ G(t) : |A| = k}|
and
p
h
u = ( 3 q 2 − 1)
.
q+h
But
Ωk (t) ≤
|E(t)|
. Thus,
k
we get the estimate
ZP (β, q, t) ≤
Proof of Theorem 5.1.2.
q+h
q
|E(t)|
2
q |V (t)|+ 3 (1 + u)|E(t)| .
(5.36)
Using estimate (5.36) and Table 5.1, we write the new bound (5.36)
for the partition function of the Potts model on
t
in terms of the dynamical variable
n(t),
and make similarly computations for the dual case. For that, we consider two cases of interest
separately.
The model on CDTs t: In this case, we can to write the
function in terms of volume n(t) of the triangulation as follow
1.
ZP (β, q, t) ≤
q+h
q
32 n(t)
1
2
new bound for the partition
3
q 2 n(t)+ 3 (1 + u) 2 n(t) .
Using estimate (5.37), we obtain a new upper bound for the partition function of the
(5.37)
q -state
Potts model coupled to CDTs
2
ΞN (β, µ) ≤ q 3
X
2
exp {−µ̃n(t)} = q 3 ZN (µ̃),
(5.38)
t
q+h
q
1
3
ln q − ln(1 + u) and ZN (µ̃) is the partition function for
2
2
pure CDTs (dened in (2.4)) in the cylinder CN with periodical spatial boundary conditions
and for the value of the cosmological constant µ̃. Hence, the inequality
where
3
µ̃ = µ − ln
2
−
3
3
eβ − 1
β
2/3
µ > ln q + e − 1 + ln 2 − ln q + ln 1 + (q − 1)
2
2
q + eβ − 1
provides a sucient condition for subcriticality of the
q -state
(5.39)
Potts model coupled to CDTs
5.4
HIGH-T EXPANSION OF THE POTTS MODEL AND PROOF OF THEOREM 5.1.2
(summation is over all Lorentzian triangulation
61
t).
Comparing the new upper bound (5.39) with bounds show in Tables 5.2 and 5.3 for the
model on CDTs, we observe which the condition (5.39) is better than conditions show in
Tables 5.2 and 5.3 for subcriticality behavior of model. Thus, using High-T expansion for
q -state
Potts model we get to obtained a better approximation of the critical curve.
Using the duality relation proved in Theorem 5.1.1 and bound (5.39), we obtain a new
condition for subcriticality of the Potts model coupled to dual CDTs
µ
3
3
q
3
3
q 2/3 − 1
β∗
>
ln e − 1 + ln 2 + ln q + β ∗
− ln q + ln 1 +
2
2
e −1
2
2
eβ ∗
3
q 2/3 − 1
3 ∗
β + ln 2 + ln 1 +
>
2
2
eβ ∗
∗
(5.40)
We will see that this same approach on dual triangulations does not improve the curves
obtained.
The model on dual CDTs t∗ : Similarly as Eq. (5.37), using Table 5.1 we get the estimate
∗
dual triangulation t
2.
on a
∗
∗
ZP (β , q, t ) ≤
q+h
q
32 n(t)
3
2
q n(t)+ 3 (1 + u) 2 n(t) .
Using (5.41), we obtain an upper bound for the partition function of the
(5.41)
q -state Potts model
coupled to dual CDTs
2
Ξ∗N (β ∗ , µ∗ ) ≤ q 3
X
2
exp {−µ̃n(t)} = q 3 ZN (µ̂),
(5.42)
t
q+h
3
where
− ln q − ln(1 + u) and ZN (µ̂) is the partition function for
q
2
pure CDTs in the cylinder CN with periodical spatial boundary conditions and for the value
of the cosmological constant µ̂. Hence, we obtain the inequality
3
µ̂ = µ − ln
2
∗
∗
3
1
3
eβ − 1
β∗
2/3
µ > ln q + e − 1 + ln 2 − ln q + ln 1 + (q − 1)
,
2
2
2
q + eβ ∗ − 1
∗
(5.43)
that provide a sucient condition for subcriticality of the Potts model coupled to dual CDTs
(summation is over all dual Lorentzian triangulation
t∗ )
Finally, using the duality relation proved in Theorem 5.1.1 and bound (5.43), we obtain
62
5.5
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
a new condition for subcriticality of the Potts model coupled to CDTs
3
3
q
3
3
q 2/3 − 1
β
µ >
ln e − 1 + ln 2 + ln q + β
− ln q + ln 1 +
2
2
e −1
2
2
eβ
3
q 2/3 − 1
3
β + ln 2 + ln 1 +
.
>
2
2
eβ
(5.44)
It is easy see that (5.43) and (5.44) does not improve the curves obtained in (5.40) and
(5.39), respectively.
Convergence of the Gibbs measure
Pβ,µ
N
follows as a corollary. This concludes the proof.
5.5 Connection between transfer matrix and FK representation
In this section, we nd a connection between transfer matrix approach and FK representation for the Ising model coupled to CDTs, comparing the curve obtained by transfer
matrix approach and the curves obtained by FK representation. In this section we work with
Potts model coupled to DUAL CDTs and will use notations of before chapters.
5.5.1
q = 2 (Ising) systems
The transfer matrix method provides a curve
µ∗ = ψ(β ∗ ) (blue line in Figure 5.5), dened
in (4.21), that satises
dψ +
(0 ) = 0.
dβ ∗
Therefore, we expect that critical curve satised the same property. Additionally, conditions
< ∞ generate curves µ∗ = γN (β ∗ ) in the quadrant of parameters β ∗ , µ∗ , such
N
(0+ ) = 0 for all N (see Proposition 4.4.3 and 4.4.4 in Chapter 4).
≤ γN and dγ
dβ ∗
N
tr(K )
γN +1
that
We dene the functions
3
β∗
ϕinf (β ) = max 2 ln 2, ln e − 1 + ln 2 ,
2
∗
and
∗ 3
2/3
β∗
ϕsup (β ) = min ψ(β ), ln 2 + e − 1 + ln 2 .
2
∗
Graphs of function
ϕinf
is a lower barrier for the graph of
curve of the model. Further, graphs of function
ϕsup
γN
for all
N,
and the critical
provides a better upper bound for the
critical curve of Ising model coupled to dual CDTs. Furthermore, in low temperature, the
5.5
63
CONNECTION BETWEEN TRANSFER MATRIX AND FK REPRESENTATION
free energy satisfy
1
3 ∗
∗
∗
∗
lim
ln ΞN (β , µ ) ≈ ln Λ µ − β .
N →∞ N
2
Therefore, maximal eigenvalue
Λ
of operator
K
can be approximated by
3 ∗
∗
Λ(β , µ ) ≈ Λ µ − β ,
2
∗
where
Λ
∗
is dened in Chapter 2 in Eq. (2.10).
8
6
4
2
0
-2
1
Figure 5.5:
5.5.2
2
3
4
Region where the critical curve of the Ising model coupled to dual CDTs can be located.
q -Potts systems
As in Ising model case, (See Chapter 3), the transfer-matrix formalism suggests rewrite
the partition function as
ΞN (β, µ) = tr KN .
where we assume periodic spatial boundary condition and the operator
(5.45)
K
is dened by
o
n µ
K((t, σ), (t0 , σ 0 )) = 1t∼t0 exp − (n(t) + n(t0 ))
2
n β
o
× exp − h(σ) + h(σ 0 ) − βv(σ, σ 0 ) .
2
(5.46)
Theorem 3.2.1 and Proposition 3.2.1 given conditions for existence of Gibbs measures
for the model in terms of the trace of operator
Estimating tr(KKT ):
tr(KKT ) < ∞ guarantees that K and KT are HilbertT
2
T 2
the operators K and K are bounded and K and (K )
The condition
Schmidt operators. Consequently,
K.
64
5.5
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
are of trace class. In particular,
K2
and
(KT )2
belong to space
Cp
for all
p ≥ 2 (see Appendix
A).
By denition the trace (A.2.1), we need to calculate the series
X
tr(KT K) =
K 2 ((t, σ), (t0 , σ 0 )).
(5.47)
(t,σ ),(t0 ,σ 0 )
As in Chapter 3, we represent the triangulation
σ
t
and its supported spin-conguration
as
t := (tup , tdo )
and
σ := (σ up , σ do ).
Here
tup = (t1up , . . . , tnup ), tdo = (t1do , . . . , tm
do ),
and
1
n
1
m
, . . . , σup
), σ do = (σdo
, . . . , σdo
),
σ up = (σup
assuming that the supporting single-strip triangulation
t
contains
n
up-triangles and
m
down-triangles. (The actual order of up- and down-triangles and supported spins does not
matter.)
(t0 , σ 0 ) (see proof of Lemma 3.2.1). Let recall that the
0
consistent (t ∼ t ) i number of the down-triangles in t equals
The same can be done for the pair
triangulations
t
t0
and
that of up-triangles in
are
t0 .
(t0 , σ 0 ) into a summation over
(t0up , σ 0up ) and (t0do , σ 0do ). Firstly, x a pair (t0up , σ 0up ) and make the sum over (t0do , σ 0do ). Note
0
0
0
that the term V ((t, σ), (t , σ )) depends only on σ do and σ up . Consequently,
To calculate the sum (5.47) we divide the summation over
X
t0do ,
σ
K 2 ((t, σ), (t0 , σ 0 ))
(5.48)
0
do
= e−βH(σ ) e−2βV ((t,σ ),(t ,σ )) e−µn(t)
0
0
X
e−βH(σ ) e−µn(t ) .
0
0
(t0do ,σ 0do )
The sum in the right-hand side of (5.48) can be represented in a matrix form. Denote by
the unit vectors in
q
T
R : ek = (0, . . . , 1, . . . , 0)
. Next, let us introduce a
q×q
matrix
T
ek
where
5.5
CONNECTION BETWEEN TRANSFER MATRIX AND FK REPRESENTATION



 β
1 ··· 1 
e








β
 1 e ··· 1 



T = e−µ 



. .
.
 . . ... . 
.
.
.










β
1 1 ··· e
(5.49)
n(i), i = 1, . . . , nup (t0 ) the number of down-triangles
(i + 1)th up-triangles in t0 . Let nup (t0 ) = k then
Denote by
ith
and
e−βH(σ ) e−µn(t ) =
0
X
0
up
M
t0
which are between the
T n(l)+1 eσ0 l+1
eT
σ0 l
up
P
i
n(i)≥1 l=1
k Y
T
T
e
M
e
eT
−
e
l+1
l+1
σ 0 up
σ 0 up
σ0 l
σ0 l
(5.50)
up
l=1
where the matrix
in
up
n(i)≥0:
k Y
k Y
X
t0do ,σ 0do
=
65
l=1
is the sum of the geometric progression
M=
∞
X
Tn
(5.51)
n=1
Using the same procedure we can obtain the sum over all up-triangles into the triangulation
t. The only dierence is
nup (t0 ) = ndo (t) = k then
the existence of marked up-triangle in the strip: let as before
−βH(σ ) −µn(t)
X
e
e
=
tup ,σ up
k−1
Y
eT
l M eσ l+1
σup
up
T
2
1
eσup
k M eσup
(5.52)
l=1
Supposing the existence of the matrix
X
M
and using (5.50) and (5.52) we obtain the following:
K 2 ((t, σ), (t0 , σ 0 )) = e−2βV ((tdo ,σ do ),(tup ,σ up ))
0
X
0
tup ,σ up t0do ,σ 0do
e−βH(σ ) e−µn(t)
X
×
tup ,σ up
X
e−βH(σ ) e−µn(t )
0
0
(t0do ,σ 0do )
= e−2βV ((tdo ,σ do ),(tup ,σ up ))
k hY
k−1
Y
T
T
T
2
1
M
e
e
M
e
×
eσ0 l M eσ0 l+1
e
l+1
l
k
σ
σup
σ
σ
up
up
0
0
up
l=1
k Y
eT
T eσ0 l+1
σ0 l
up
−
do
l=1
k−1
Y
up
l=1
eT
σ l M eσ l+1
do
l=1
do
do
i
2
1
eT
M
e
.
k
σup
σup
(5.53)
66
5.5
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Necessary and sucient condition for the convergence of the matrix series for
the maximal eigenvalue of matrix
T
is less then 1. The eigenvalues of
λ1 = e−µ (q + eβ − 1),
and the above condition means that
λ1 < 1
T
M
are
λ2 = e−µ (eβ − 1),
(5.54)
or, equivalently,
µ > ln(q + eβ − 1).
Under this condition (5.55), the matrix
M
(5.55)
is calculated explicitly:

M = g(β, µ)
is that

1 + f (β, µ) · · ·

1








.
.


.
×
.
..
.

.
.








1
· · · 1 + f (β, µ)
,
(5.56)
q×q
where
g(β, µ) =
e−µ
(1 − e−µ (eβ − 1))(1 − e−µ (q + eβ − 1))
and
f (β, µ) = (eβ − 1)(1 − e−µ (q + eβ − 1)).
Now, we express the above sum (5.47) as the partition function of a one-dimensional
Potts model where states are pairs of spins
T
l
l
(σdo
, σup
)
q -state
and the interaction is via the matrix
M between neighboring pairs. More
set {(i, j) : 1 ≤ i, j ≤ q}, we dene the
between the members of the pair and via matrix
precisely, introducing the lexicographic order in the
following
q2 × q2
matrices:
A=






e2β (eTn M el ) · (eTm M ek ) , n = m and l = k






(eTn M el ) · (eTm M ek )










 eβ (eTn M el ) · (eTm M ek )
, n 6= m and l 6= k
,
either
n = m or l = k,
(5.57)
5.5
67
CONNECTION BETWEEN TRANSFER MATRIX AND FK REPRESENTATION
Am =






e2β (eTn M el ) · (eTm M 2 ek ) , n = m and l = k






(eTn M el ) · (eTm M 2 ek )










 eβ (eTn M el ) · (eTm M 2 ek )
At =
Atm =
(5.58)
, n 6= m and l 6= k
,
either
n = m or l = k,






e2β (eTn T el ) · (eTm M ek ) , n = m and l = k






(eTn T el ) · (eTm M ek )










 eβ (eTn T el ) · (eTm M ek )
(5.59)
, n 6= m and l 6= k
,
either
n = m or l = k,






e2β (eTn T el ) · (eTm M 2 ek ) , n = m and l = k






(eTn T el ) · (eTm M 2 ek )










 eβ (eTn T el ) · (eTm M 2 ek )
(5.60)
, n 6= m and l 6= k
,
either
n = m or l = k.
Now for the sum under consideration (5.47) we obtain using representation (5.53)
T
tr(K
X
K) =
0
2
0
∞
X
K ((t, σ), (t , σ )) = tr
(t,σ ),(t0 ,σ 0 )
k
A
∞
X
Am − tr
k=0
k=1
At elementwise. Thus the eigenvalue of
matrix A is greater than the eigenvalue of the matrix Qt (it follows from the Perron-Frobenius
By the construction the matrix
A
Akt Atm .
is greater then
theorem). Therefore the necessary and sucient condition for the convergence in (5.47) is
A
is less than 1. In general, it is impossible to calculate its
eigenvalue analytically. For case
q = 2 (Ising model) was possible calculated its eigenvalue
q > 2 it very dicult. In the case of q = 4, we make a
that the largest eigenvalue of
(see Chapter 3 for review), but
comparison between curves obtained by duality relation and high-T expansion in Section
5.4, and curve obtained using numerical simulation. See below graph.
Finally as byproduct of Theorem 5.1.2, we have the following assertions for the free
energy for
q -state
Corollary 5.5.1.
Potts model.
The free energy for q -state Potts model coupled to dual CDTs satisfy
1
3 ∗
∗
∗
∗
∗
lim
ln ΞN (β , µ ) ≤ ln Λ µ − β − r(β ) ,
N →∞ N
2
68
5.6
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
The blue line
is the simulation of ||A||2 = 1 for q = 4. Black line: µ∗ = 3 ln 2. Green
∗
∗
line: µ∗ = 23 ln eβ − 1 + ln 2. Red line: µ∗ = 32 ln 42/3 + eβ − 1 + ln 2.
Figure 5.6:
q 2/3 − 1
3
. Moreover, in low temperature, we have that
where r(β ) = ln 1 +
2
eβ ∗
∗
3 ∗
1
∗
∗
∗
ln ΞN (β , µ ) ≈ ln Λ µ − β .
lim
N →∞ N
2
Therefore, maximal eigenvalue Λ of operator K can be approximated by
3 ∗
∗
Λ(β , µ ) ≈ Λ µ − β ,
2
∗
∗
where Λ is dened in Chapter 2 in Eq. (2.10).
5.6 Discussion and outlook
This chapter we dene and study a Potts model coupled to CDTs and dual CDTs employing FK representation and duality on graphs. The results obtained serve for the Ising model
(see Section 5.5.1). In particular, we nd a better region where the free energy there exists
and can be extended analytically (line
analyticity of maximal eigenvalue
Λ
I
and
I0
of operator
in Figure 5.1), and this region depend on
K
and eigenvalue
Λ,
dened in Eqn (2.10).
This remark permite us to give the conjecture that the boundary of the critical domain
coincides with the locus of points (β ,
µ)
where
Λ
loses either the property of positivity or
the property of being a simple eigenvalue.
(β, µ) satisfy hypothesis of Theorem 5.1.2, we don't have information on
either K belong to Cp or not, for some p > 2. This issue needs a further study.
We can give tha follos assertion: If (β, µ) satisfy the subcritical behavior of Theorem 5.1.2,
β,µ
the limiting probability distribution P
is represented by a positive recurrent Markov chain
Notice that, if
DISCUSSION AND OUTLOOK
with states
(t, σ)
69
as Theorem 3.2.2. In the subcritical region of Theorem 5.1.2, the typical
triangulation for annealed Potts model coupled to CDTs is the same as subcritical case in
pure CDT (see Theorem 2.2.2).
The new bounds for the critival curve for arbitrary
q
suggests that the Potts model
coupled to CDTs exhibit a phase transition only on the critical curve and a rst-order
singularity at a unique point
βcr ∈ (0, ∞).
Additionally, the triangulations in the annealed
model exhibit critical behavior as critical case in pure CDT (see Theorem 2.2.2) only on the
critical curve. This direction also requires further research.
70
POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION
Appendix A
The von Neumann-Schatten Classes of
Operators
This appendix in concerned with certain classes
on a Hilbert space
H.
Cp (1 ≤ p < ∞)
of linear operators.
These classes are important for to study the transfer operator in
statistical mechanics because that operator encodes information and to study the behavior
of the statistical mechanics system.
A.1 The space Cp and rst properties
Cp
In this section we dene the space
and given some properties.
Denition A.1.1.
When 1 ≤ p < ∞, Cp is the set of all operators T in B(H) which satisfy
the following condition: for each orthonormal system {φk : k ∈ K} in H,
X
|hT φk , φk i|p < ∞.
k∈K
We shall adopt the convention that
Cp ⊆ Cq if 1 ≤ p ≤ q ≤ ∞. We
T ∈ Cp , and that, if 1 ≤ p < ∞, then
and
C∞
is
B(H).
can to see that
each element of
Cp is a linear subspace of B(H)
T ∗ ∈ Cp (adjoint operator) whenever
Cp is a compact operator.
Each
Lemma A.1.1.
Suppose that 1 ≤ p < ∞, T is a compact self-adjoint operator on H, and
{λn } is the sequence of non-zero eigenvalues of T , counted according to their multiplicities.
1. If T ∈ Cp , then
2. If
P
n
P
n
|λn |p < ∞.
|λn |p < ∞, then T ∈ Cp and, for each orthonormal system {φk : k ∈ K} in H,
X
|hT φk , φk i|p <
X
n
k∈K
71
|λn |p .
72
APPENDIX A
A.2 The trace class C1
In this section we will dene the trace of a operator on the class
Lemma A.2.1.
C1 .
Let T ∈ C1 and suppose that {φk : k ∈ K} is an orthonormal basis in H.
P
1. The sum k hT φk , φk i exist, and does not depend on the particular choise of the orthonormal basis {φk : k ∈ K}.
P
P
2. If T = T ∗ , then k hT φk , φk i = k λk , where {λk } is the sequence of non-zero eigenvalues of T , counted according to their multiplicities.
Denition A.2.1.
The ideal C1 in B(H) is called trace class of operators on H. If T ∈ C1
and {φk : k ∈ K} is an orthonormal basis in H, then the trace of T , denoted by tr(T ), is
dened by the equation
X
tr(T ) =
hT φk , φk i.
k
Lemma A.2.1 shows that tr(T ) depends only on
T
basis), and that tr(T ) is the sum of the eigenvalues of
(not on the choice of the orthonormal
T
when
T = T ∗.
The main algebraic properties of tr are the following.
Theorem A.2.1.
Suppose that S, T ∈ C1 , A ∈ B(H) and α, β are scalars.
1. tr(αS + βT ) = αtr(S) + β tr(T ).
2. tr(S ∗ ) = tr(S).
3. tr(S) > 0 if S > 0.
4. tr(AS) = tr(SA).
The main result of this section, used in Chapter 3, is the following.
Theorem A.2.2.
Suppose that T is a trace class operator acting on a Hilbert space H,
and {λk } is the sequence of non-zero eigenvalues of T , counted according to their algebraic
multiplicities. Then
X
tr(T ) =
λk .
(A.1)
k
A.3 The Banach space Cp
Suppose that
is a compact operator acting on
∗
H, and denote by VT HT the polar decom. Remember that VT is a partial isometry
there exist a decreasing sequence {µn } of
1/2
T . Then T = VT HT and HT = (T T )
closed range RH of H . It is well know that
position of
on the
T
THE BANACH SPACE
positive real numbers (the eigenvalues of
orthonormal sequence
{φn }, {ψn },
CP
73
HT , counted according to their multiplicities), and
such that
HT (x) =
X
µn hx, φn iφn ,
n
T (x) =
X
µn hx, φn iψn .
n
Given
p≤1
the function
fp (t) = tp
is continuous on the non-negative real axis, and hence
HT . The operator
p
HT is compact and can be represented by
also on the spectrum of the positive operator
HTp . The operator
HTp (x) =
X
fp (HT )
will be denoted by
µpn hx, φn iφn .
n
Lemma A.3.1. Suppose 1 ≤ q ≤ p < ∞ and T ∈ B(H). Then the following three conditions
are equivalent
1. T ∈ Cp ,
2. HT ∈ Cp ,
p/q
3. HT
∈ Cq .
From Lemma A.3.1 and the equivalence of conditions
it follows that a compact operator
eigenvalues of
HT = (T ∗ T )1/2
Denition A.3.1.
T
satises
on
P
H lies Cp if
p
n µn < ∞ .
(i)
and
(ii)
in the before lemma,
only if the sequence
{µn }
of non-zero
Suppose 1 ≤ p < ∞ and T ∈ Cp . Then, we dene
!1/p
X
||T ||p = [tr(HTp )]1/p =
µpn
.
n
It is not immediately obvious that
|| · ||p
is a norm on
Cp .
Since
||T || = ||HT || = µ1
(maximal eigenvalue), we have
||T || ≤ ||T ||p ,
for
Lemma A.3.2.
For each T ∈ C1 , |tr(T )| ≤ ||T ||1 .
Lemma A.3.3.
For each T ∈ Cp , ||T ∗ ||p = ||T ||.
T ∈ Cp .
Lemma A.3.4.
Suppose 1 ≤ p < ∞, T ∈ Cp , and {λn } is the sequence of non-zero eigenvalues of T , counted according to their algebraic multiplicities. Then
!1/p
X
n
|λn |p
≤ ||T ||p .
74
APPENDIX A
A.4 The Hilbert-Schmidt class
Denition A.4.1.
The ideal C2 in B(H) is called the Hilbert-Schmidt class of operators on
H.
If
VT HT
is the polar decomposition of an element
T
of
C2 ,
then
(tr(T ∗ T ))1/2 = (tr(HT2 ))1/2 = ||T ||2 .
Theorem A.4.1.
Suppose that T ∈ B(H), {φk } and {ψk } are orthonormal bases in H. The
the following three conditions are equivalent.
1.
P
2.
P
k
||T φk ||2 < ∞,
j,k
|hT φj , ψk i|2 < ∞,
3. T ∈ C2 .
When these conditions are satised, the sums occurring in (i) and (ii) are both equal to
||T ||22 .
Appendix B
Krein-Rutman theorem
The Krein-Rutman theorem plays a very important role in linear fuctional analysis, as
it provides the abstract basis for the proof of the existence of various principal eigenvalues,
which in turn are crucial in transfer matrix formalism of statistical mechanics system (and
another areas as nonlinear partial dierential equations, bifurcation theory, etc). In this
appendix, we will give the well-known Krein-Rutman theorem.
B.1 Krein-Rutman Theorem and the Principal Eigenvalue
X a Banach space. By cone K ⊂ X we mean a closed convex set such that λK ⊂ K
for all λ ≥ 0 and K ∩ (−K) = {0}. A cone K in X induce a partial ordering ≤ by the rule:
u ≤ v if and only if v − u ∈ K . A Banach space with such an ordering is usually called
Let
a partially ordered Banach space and the cone generating the partial ordering is called the
K − K = X , i.e., the set {u − v : u, v ∈ K} is dense in X ,
then K is called a total cone. If K − K = X , K is called a reproducing cone. If a cone has
o
nonempty interior K , then it is called a solid cone. Any solid cone has the property that
K − K = X , in particular, it is total. We write u > v if u − v ∈ K \ {0}, and u v if
u − v ∈ K o.
positive cone of the space. If
The main results of this appendix, used in Chapter 3, are the following.
Theorem B.1.1
.
Let X a Banach space, K ⊂ X a
total cone and T : X → X a compact linear operator that is positive, i.e., T (K) ⊂ K , with
positive spectral radius r(T ). Then r(T ) is an eigenvalue with an eigenvector u ∈ K \ {0}:
T u = r(T )u. Moreover, r(T ∗ ) = r(T ) is an eigenvalue of T ∗ .
(The Krein-Rutman Theorem [KR48])
Let us now use Theorem B.1.1 to derive the following useful result.
Theorem B.1.2.
Let X a Banach space, K ⊂ X a solid cone, T : X → X a compact linear
operator which is strongly positive, i.e., T u 0 if u > 0. Then
75
76
APPENDIX B
1. r(T ) > 0, and r(T ) is a simple eigenvalue with an eigenvector v ∈ K o ; ; there is no
other eigenvalue with a positive eigenvector.
2. |λ| < r(T ) for all eigenvalues λ 6= r(T ).
Let us recall that
and
n
(λI − T ) w = 0
λ
T if there
w ∈ span{v}.
is a simple eigenvalue of
for some
n≥1
implies
exists
v 6= 0
such that
T v = λv
Bibliography
[AAL99] J. Ambjørn, K. N. Anagnostopoulos e R. Loll.
A new perspective on matter
coupling in 2d quantum gravity. página 104035, 1999. 1, 2, 15, 16, 42, 44, 45
[AALP08] J. Ambjørn, K. N. Anagnostopoulos, R. Loll e I. Pushkina.
Shaken, but not
stirred - potts model coupled to quantum gravity. 2008. 1
[ADJ97] J. Ambjørn, B. Durhuus e T Jonsson.
theory approach.
Quantum geometry. A statistical eld
Cambridge Monogr. Math. Phys. Cambridge University Pres,
Cambridge, UK, 1997. 1, 41
[AGGS12] J. Ambjørn, L. Glaser, A. Görlich e Y. Sato. New multicritical matrix models
and multicritical 2d CDT. página 109, 2012. 2
[Aiz94] M. Aizenman. Geometric aspects of quantum spin states.
Mathematical Physics,
Communications in
164:1763, 1994. 3, 27, 29, 31
[AJ06] J. Ambjørn e J. Jurkiewics. The universe from scratch. páginas 103117, 2006.
1, 41
[AL98] J. Ambjørn e R. Loll.
Non-perturbative lorentzian quantum gravity, causality
and topology change. páginas 407434, 1998. 1, 7, 41
[AN72] K. B. Athreyat e P. E. Ney.
Branching processes.
Die Grundlehren der mathe-
matischen. Springer-Verlag, 1972. 11
[AS03] O. Angel e O. Schramm. Uniform innite planar triangulations. páginas 191213,
2003. 1
[AZ12a] M. R. Atkin e S. Zohren. An analytical analysis of CDT coupled to dimer-like
matter. página 445, 2012. 2
[AZ12b] M. R. Atkin e S Zohren. On the quantum geometry of multi-critical cdt. página
037, 2012. 2
[BC12] V. Beara e H. Duminil Copin.
The self-dual point of the two- dimensional
random-cluster model is critical for
q ≥ 1.
páginas 511542, 2012. 42, 48, 49
[BDG02] J. Bouttier, P. Di Francesco e E. Guitter.
Census of planar maps: From the
one-matrix model solution to a combinatorial proof. página 477, 2002. 1, 42
[BK87] D. V Boulatov e V. A. Kazakov. The ising model on random planar lattice: The
structure of phase transition and the exact critical exponents. página 379, 1987.
2
77
78
BIBLIOGRAPHY
[BL07] D. Benedetti e R Loll. Quantum gravity and matter: Counting graphs on causal
dynamical triangulations. páginas 863898, 2007. 1, 2, 42
[BMS11] M. Bousquet-Melou e G. Schaeer. The degree distribution in bipartite planar
maps: applications to the Ising model. 2011. 2
[ES88] R. G. Edwards e A. D. Sokal. Generalization of the fortuin-kasteleyn- swendsenwang representation and monte carlo algorithm. páginas 20092012, 1988. 47,
48
[FGZJ95] P. Di Francesco, P. H. Ginsparg e J. Zinn-Justin. 2-d gravity and random matrices. páginas 1133, 1995. 1, 42
[FK72] C.M. Fortuin e R.W. Kasteleyn. On the random-cluster model i. introduction
and relation to other models. páginas 536564, 1972. 2, 27, 42, 46, 47
[GG11] J.-F. L. Gall e G.Miermont.
Scaling limits of random trees and planar maps.
2011. 1
[Gri06] G.R. Grimmett.
The random-cluster model.
Progress in Probability. Springer,
2006. 45, 47
[Her14] J. Cerda Hernández. Critical region for an ising model coupled to causal dynamical triangulations. 2014. 2, 42, 44, 46
[HYSZ13] J.C. Hernandez, A. Yambartsev, Y. Suhov e S. Zohren. Bounds on the critical
line via transfer matrix methods for an ising model coupled to causal dynamical
triangulations. página 063301, 2013. 2, 34, 40, 42, 44, 45, 46
[Iof09] D. Ioe. Stochastic geometry of classical and quantum ising models. 2009. 3, 27,
29, 31
[JAZ07] W. Westra J. Ambjørn, R. Loll e S. Zohren. Putting a cap on causality violations
in CDT. página 017, 2007. 2
[JAZ08a] Y. Watabiki W. Westra J. Ambjørn, R. Loll e S. Zohren. A causal alternative
for
c=0
strings. página 3355, 2008. 2
[JAZ08b] Y. Watabiki W. Westra J. Ambjørn, R. Loll e S. Zohren. A matrix model for 2D
quantum gravity dened by causal dynamical triangulations. páginas 252256,
2008. 2
[JAZ08c] Y. Watabiki W. Westra J. Ambjorn, R. Loll e S. Zohren. A new continuum limit
of matrix models. páginas 224230, 2008. 2
[JAZ08d] Y. Watabiki W. Westra J. Ambjørn, R. Loll e S. Zohren. A string eld theory
based on causal dynamical triangulations. página 032, 2008. 2
[Kaz86] V. A. Kazakov. Ising model on a dynamical planar random lattice: Exact solution.
páginas 140144, 1986. 2
[KR48] M. G. Krein e M. A. Rutman.
Linear operators leaving invariant a cone in a
banach space. páginas 395, 1948. iii, v, 3, 10, 18, 75
BIBLIOGRAPHY
79
[KY12] M. Krikun e A. Yambartsev. Phase transition for the Ising model on the critical
Lorentzian triangulation. páginas 422439, 2012. 10
[MAC92] Aizenman M., Klein A. e Newman C.M.
Percolation methods for disordered
quantum ising model. páginas 129137, 1992. 3
[MYZ01] V. Malyshev, A. Yambartsev e A. Zamyatin. Two-dimensional lorentzian models.
páginas 118, 2001. 7, 8, 10, 45, 54
[Ons44] L. Onsager. Crystal statistics. i. a two-dimensional model with an order-disorder
transition. páginas 117149, 1944. 2
[Rin71] J.R. Ringrose.
Compact Non-Self-Adjoint Operators.
Van Nostrand Reinhold
Co., 1971. 17
[Sch97] G. Schaeer. Bijective census and random generation of Eulerian planar maps
with prescribed vertex degrees. página R20, 1997. 1, 42
[SYZ13] V. Sisko, A. Yambartsev e S. Zohren. A note on weak convergence results for
uniform innite causal triangulations. 2013. 5, 10
[Tut62] W. T. Tutte. A census of planar triangulations. páginas 2138, 1962. 1, 41
[Tut63] W. T. Tutte. A census of planar maps. páginas 249271, 1963. 1, 41