Asymptotic expansion of β matrix models in the multi-cut regime
G. Borot 1 , A. Guionnet 2
1
Section de Mathématiques, Université de Genève
2-4 rue du Lièvre, Case Postale 64, 1211 Genève 4, Switzerland.
2
UMPA, CNRS UMR 5669, ENS Lyon,
46 allée d’Italie, 69007 Lyon, France.
&
Department of Mathematics, MIT,
77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA.
Abstract
We push further our study of the all-order asymptotic expansion in β matrix models with a
confining, offcritical potential, in the regime where the support of the equilibrium measure is a reunion
of segments. We first address the case where the filling fractions of those segments are fixed, and show
the existence of a 1{N expansion to all orders. Then, we study the asymptotic of the sum over filling
fractions, in order to obtain the full asymptotic expansion for the initial problem in the multi-cut
regime. We describe the application of our results to study the all-order small dispersion asymptotics
of solutions of the Toda chain related to the one hermitian matrix model (β “ 2) as well as orthogonal
polynomials outside the bulk.
1 gaetan.borot@unige.fr
2 guionnet@math.mit.edu
1
Introduction
This paper deals with the all-order asymptotic expansion for the partition function and multilinear
statistics of β matrix models. These laws represent a generalization of the joint distribution of the
N eigenvalues of the Gaussian Unitary Ensemble [Meh04]. The convergence of the empirical measure
of the eigenvalues is well-known (see e.g. [dMPS95]), and we are interested in the all-order finite size
corrections to the moments of this empirical measure. This problem has received a lot of attention in
the regime when the eigenvalues condensate on a single segment, usually called the one-cut regime. In
this case, a central limit theorem for linear statistics has been proved by Johansson [Joh98], while a full
1{N expansion was derived first for β “ 2 [APS01, EM03], then for any β ą 0 in [BG11]. On the other
hand, the multi-cut regime remained poorly understood at a rigorous level until recently, except for
β “ 2 which is related to integrable systems, and can be treated with the powerful asymptotic analysis
techniques for Riemann-Hilbert problems, see e.g. [DKM` 99b]. Nevertheless, a heuristic derivation
of the asymptotic expansion for the multi-cut regime was proposed to leading order by Bonnet, David
and Eynard [BDE00], and extended to all orders in [Eyn09], in terms of Theta functions and their
derivatives. It features oscillatory behavior, whose origin lies in the tunneling of eigenvalues between
the different connected components of the support. These heuristics, initially written for β “ 2,
trivially extend to β ą 0, see e.g. [Bor11].
Lately, M. Shcherbina has established this asymptotic expansion up to terms of order 1 [Shc11,
Shc12]. This allows for instance the observation that linear statistics do not always satisfy a central
limit theorem (this fact was already noticed for β “ 2 in [Pas06]). In this paper, we go beyond the
Op1q and put the heuristics of [Eyn09] to all orders on a firm mathematical ground. As a consequence
for β “ 2, we can establish the full asymptotic expansion outside of the bulk for the orthogonal
polynomials with real-analytic potentials, and the all-order asymptotic expansion of certain solutions
of the Toda lattice in the continuum limit. The same method would allow to justify rigorously the
asymptotics of skew-orthogonal polynomials (β “ 1 and 4) outside of the bulk, derived heuristically
in [Eyn01].
1.1
Definitions
We consider the probability measure µVN,β on BN given by:
;B
dµVN,β
pλq “
1
N
ź
V ;B
ZN,β
i“1
dλi 1B pλi q e´
βN
2
V pλi q
ź
|λi ´ λj |β .
(1.1)
1ďiăjďN
V ;B
B is a reunion of closed intervals of RYt˘8u, β is a positive number, and ZN,β
is the partition function
so that (1.1) has total mass 1. This model is usually called the β ensemble [Meh04, DE02, For10]. We
introduce the unnormalized empirical measure MN of the eigenvalues:
MN “
N
ÿ
δλi ,
(1.2)
i“1
and we consider several types of statistics for MN . We sometimes denote Λ “ diagpλ1 , . . . , λN q.
Correlators
We introduce the Stieltjes transform of the n-th order moments of the empirical measure, called
disconnected correlators:
ˆ
”´ ˆ dM pξ q
dMN pξn q ¯ı
N 1
Ăn px1 , . . . , xn q “ µV ;B
W
¨
¨
¨
.
(1.3)
N,β
x1 ´ ξ1
xn ´ ξn
1
They are holomorphic functions of xi P CzB. For reasons related to concentration of measures, it is
more convenient to consider the correlators to study large N asymptotics:
řn
ti
´
¯ˇ
2
V ´ βN
i“1 xi ´‚ ;B ˇ
Wn px1 , . . . , xn q “ Bt1 ¨ ¨ ¨ Btn ln ZN,β
ˇ
ti “0
;B
“ µVN,β
n
”ź
i“1
1 ı
Tr
.
xj ´ Λ c
(1.4)
By construction, the coefficients of their expansion as a Laurent series in the variable xi Ñ 8 give the
n-th order cumulants of MN . If I is a set, we introduce the notation xI “ pxi qiPI for a set of variables
indexed by I. The two type of correlators are related by:
n
ÿ
Ăn px1 , . . . , xn q “
W
s
ź
ÿ
W|Ji | pxJi q.
(1.5)
s“1 J1 Y¨¨¨
9 YJ
9 s “I i“1
If ϕn is an analytic function in n variables in a neighborhood of Bn , the n-linear statistics can be
deduced as contour integrals of the disconnected correlators:
;B
µVN,β
”
N
ÿ
˛
ı
ϕn pλi1 , . . . , λin q “
i1 ,...,in “1
B
dξ1
¨¨¨
2iπ
˛
B
dξn
Ăn pξ1 , . . . , ξn q.
ϕn pξ1 , . . . , ξn q W
2iπ
(1.6)
We remark that the knowledge of the correlators for a smooth family of potentials pVt qt determines
the partition function up to an integration constant, since:
Vt ;B
Bt ln ZN,β
“´
˛
N
ı
βN Vt ;B ” ÿ
βN
dξ
µN,β
Bt Vt pλi q “ ´
Bt Vt pξq W1 pξq
2
2
2iπ
B
i“1
(1.7)
Kernels
Let c be a n-uple of non zero complex numbers. We introduce the n-kernels:
«
ff
n
ź
V ;B
cj
Kn,c px1 , . . . , xn q “ µN,β
det pxj ´ Λq
j“1
řn
“
V´ 2
ZN,β βN
j“1
cj lnpxj ´‚q;B
V ;B
ZN,β
.
(1.8)
When cj are integers, the kernels are holomorphic functions of xj P CzB. When cj are not integers, the
kernels are multivalued holomorphic functions of xj in CzB, with monodromies around the connected
components of B and around 8.
In particular, for β “ 2, K1,1 pxq is the monic N -th orthogonal polynomial associated to the weight
1B pxq e´N V pxq dx on the real line, and K2,p1,´1q px, yq is the N -th Christoffel-Darboux kernel associated
to those orthogonal polynomials, see Section 2.
1.2
Equilibrium measure and multi-cut regime
By standard results of potential theory, see [Joh98] or the textbooks [Dei99, Theorem 6] or [AGZ10,
Theorem 2.6.1 and Corollary 2.6.3], we have:
Theorem 1.1 Assume that V : B Ñ R is a continuous function, and if τ 8 P B, assume that:
lim inf
xÑτ 8
V pxq
ą 1.
2 ln |x|
2
(1.9)
If V depends on N , assume also that V Ñ V t0u in the space of continuous function over B for the
sup norm. Then, the normalized empirical measure LN “ N ´1 MN converges almost surely and in
expectation towards the unique probability measure µeq :“ µVeq;B on B which minimizes:
ˆ
¨
Erµs “ dµpξqV t0u pξq ´
dµpξqdµpηq ln |ξ ´ η|.
(1.10)
µeq has compact support, denoted S. It is characterized by the existence of a constant C such that:
ˆ
@x P B,
2 dµeq pξq ln |x ´ ξ| ´ V t0u pxq ď C,
(1.11)
B
with equality realized µeq almost surely. Moreover, if V t0u is real-analytic in a neighborhood of B, the
support consists of a finite disjoint union of segments:
S“
g
ď
Sh “ rαh´ , αh` s,
Sh ,
(1.12)
h“0
µeq has a density of the form:
g
`
´
dµeq
Spxq ź `
“
pαh ´ xqρh {2 px ´ αh´ qρh {2 ,
dx
π h“0
(1.13)
where ρ‚h is `1 (resp. ´1) if the corresponding edge is soft (resp. hard), and S is analytic in a
neighborhood of S.
The goal of this article is to establish an all-order expansion of the partition function, the correlators
and the kernels, in all such situations.
1.3
Assumptions
We will refer throughout the text to the following set of assumptions.
Hypothesis 1.1
‚ (Regularity) V : B Ñ R is continuous, and if V depends on N , it has a limit V t0u in the space
of continuous functions over rb´ , b` s for the sup norm.
‚ (Confinement) If τ 8 P B, lim inf xÑτ 8
V pxq
2 ln |x|
ą 1.
‚ (g ` 1-cut regime) The support of µVeq;B is of the form S “
αh´ ă αh` .
Ťg
h“0
Sh where Sh “ rαh´ , αh` s with
‚ (Control of large deviations) The effective potential U V ;B pxq “ V pxq ´ 2
achieves its minimum value on S only.
´
ln |x ´ ξ|dµVeq;B pξq
‚ (Offcriticality) In the equilibrium measure (1.13), Spxq ą 0 in S.
At some point, we shall need to add a stronger assumption concerning off-criticality:
Hypothesis 1.2 The same as Hypothesis 1.3, and:
‚ (Strong off-criticality) For any soft edge αh‚ , we have S 1 pαh‚ q ‰ 0.
3
We will also require regularity of the potential:
Hypothesis 1.3
‚ (Analyticity) V extends as a holomorphic function in some open neighborhood U of S.
‚ (1{N expansion of the potential) There exists a sequence pV tku qkě0 of holomorphic functions in
U and constants pv tku qkě0 such that, for any K ě 0,
K
ˇ
ˇ
ÿ
ˇ
ˇ
N ´k V tku pξqˇ ď v tKu N ´pK`1q .
sup ˇV pξq ´
ξPU
(1.14)
k“0
In Section 6, we shall weaken Hypothesis 1.3 by allowing complex perturbations of order 1{N and
harmonic functions instead of analytic functions:
Hypothesis 1.4 V : B Ñ C can be decomposed as V “ V1 ` V2 where:
‚ For j “ 1, 2, Vj extends to a holomorphic function in some neighborhood U of B. There exists a
tku
tku
sequence of holomorphic functions pVj qkě0 and constants pvj qkě0 so that, for any K ě 0:
K
ˇ
ˇ
ÿ
ˇ
ˇ
tku
tKu
sup ˇVj pξq ´
N ´k Vj pξqˇ ď vj N ´pK`1q .
ξPU
t0u
‚ V t0u “ V1
t0u
` V2
(1.15)
k“0
is real-valued on B.
The topology for which we study the large N expansion of the correlators is described in § 5.1, and
amounts to controlling the (moments of order m)ˆC m uniformly in m for some constant C ą 0. We
now describe our strategy and announce our results.
1.4
Main result with fixed filling fractions
Before coming to the multi-cut regime, we analyze a different model where the number of λ’s in a
Ťg
`
small enlargment of Sh is fixed. Let A “ h“0 Ah where Ah “ ra´
h , ah s are pairwise disjoint segments
´
`
`
such that a´
h ă αh ă αh ă ah . We introduce the set:
!
Eg “ Ps0, 1rg ,
g
ÿ
)
h ă 1 .
(1.16)
h“1
řg
If P Eg , we denote 0 “ 1 ´ h“1 h , we let N “ ptN 0 u, tN 1 u, . . . , tN g uq, and consider the
śg
h
probability measure on h“0 AN
h :
;A
dµVN,,β
pλq
“
1
g ”ź
Nh
ź
V ;A
ZN,,β
h“0
ź
ˆ
dλh,i 1Ah pλh,i q e´
βN
2
i“1
ź
V pλh,i q
ź
|λh,i ´ λh,j |β
ı
1ďiăjďN
β
|λh,i ´ λh1 ,j | .
(1.17)
0ďhăh1 ďg 1ďiďNh
1ďjďNh1
The empirical measure MN, and the correlators Wn, px1 , . . . , xn q for this model are defined as in
§ 1.1. We call h the filling fraction of Ah . It follows from the definitions that:
˛
dξ
Wn, pξ, x2 , . . . , xn q “ δn,1 N h .
(1.18)
Ah 2iπ
We will refer to (1.1) as the initial model, and to (1.17) as the model with fixed filling fractions.
Standard results from potential theory imply:
4
Theorem 1.2 Assume V regular and confining on A. Then, the normalized empirical measure
N ´1 MN, converges almost surely and in expectation towards the unique probability measure µeq,
on A which minimizes:
˙
¨ ˆ
1 t0u
Erµs “
pV pξq ` V t0u pηqq ´ ln |ξ ´ η| dµpξqdµpηq .
(1.19)
2
among probability measures with partial masses µrAh s “ h . They are characterized by the existence
of constants C,h such that:
ˆ
@x P Ah ,
2 dµeq, pξq ln |x ´ ξ| ´ V t0u pxq ď C,h ,
(1.20)
B
with equality realized µeq, almost surely. µeq, can be decomposed as a sum of positive measures µeq,,h
having compact support in Ah , denoted S,h . Moreover, if V t0u is real-analytic in a neighborhood of
A, S,h consists of a finite reunion of segments.
µeq appearing in Theorem 1.1 coincides with µeq,‹ for the optimal value ‹ “ pµeq rAh sq1ďhďg , and in
this case S‹ ,h is actually the segment rαh´ , αh` s. The key point is that, for close enough to ‹ , the
support S,h remains connected, and the model with fixed filling fraction enjoys a 1{N expansion.
Theorem 1.3 If V satisfies Hypotheses 1.1 and 1.4 on A, there exists t ą 0 such that, uniformly for
P Eg such that | ´ ‹ | ă t, we have an expansion for the correlators:
ÿ
tku
Wn, px1 , . . . , xn q “
N ´k Wn,
px1 , . . . , xn q ` OpN ´8 q.
(1.21)
kěn´2
Up to a fixed OpN ´K q and for a fixed n, (1.21) holds uniformly for x1 , . . . , xn in compact regions of
tku
p
CzA.
Besides, if the strong off-criticality of Hypothesis 1.2 is satisfied, Wn, are smooth functions of
close enough to ‹ .
We prove this theorem, independently of the nature soft/hard of the edges, in Section 5 with realanalytic potential (i.e. Hypothesis 1.3 instead of 1.4). The result is extended to harmonic potentials
(i.e. Hypothesis 1.4) in Section 6.2. Actually, we provide in Proposition 5.5 an explicit control of the
errors in terms of the distance of x1 , . . . , xk to A, and its proof makes clear that the expansion of the
p
correlators is not expected to be uniform for x1 , . . . , xn chosen in a compact of CzA
independently of
n and K.
We then compute in Section 7.1 the expansion of the partition function thanks to the expansion
of W1, , by a two-step interpolation preserving Hypotheses 1.2-1.4 between our potential V and a
reference situation where the partition function is exactly computable for finite N , in terms of Selberg
integrals.
Theorem 1.4 If V satisfies Hypotheses 1.2 and 1.4 on A, there exists t ą 0 such that, uniformly for
P Eg such that | ´ ‹ | ă t, we have:
´ ÿ
¯
N!
tku
V ;A
śg
ZN,,β
“ N pβ{2qN `e exp
N ´k F,β ` OpN ´8 q ,
(1.22)
h“0 pN h q!
kě´2
řg
with e “ h“0 eρ´ ,ρ` , where:
h
e`` “
h
3 ` β{2 ` 2{β
,
12
e`´ “ e´` “
β{2 ` 2{β
,
6
e´´ “
´1 ` 2{β ` β{2
,
4
(1.23)
tku
and we recall ρ‚h “ 1 for a soft edge and ρ‚h “ ´1 for a hard edge. Besides, F,β is a smooth function
t´2u
of close enough to ‹ , and at the value “ ‹ , the derivative of F,β
negative definite.
5
vanishes and its Hessian is
Up to a given OpN ´K q, all expansions are uniform with respect to parameters of the potential and
of chosen in a compact set so that the assumptions hold. The power of N in prefactor is universal
in the sense that it only depends on the nature of the edges, and its value can be extracted from the
large N expansion of Selberg type integrals. Theorems 1.3-1.4 are the generalizations to the fixed
filling fraction model of our earlier results about existence of the 1{N expansion in the one-cut regime
[BG11] (see also [Joh98, APS01, EM03, GMS07, KS10] for previous results concerning the one-cut
regime in β “ 2 or general β ensembles).
1.5
Main results in the multi-cut regime
Ťg
Let us come back to the initial model (1.1), and take A “ h“0 Ah Ď B a small enlargement of
V ;B
the support S as in the previous paragraph. It is well-known that the partition function ZN,β
can
V ;A
be replaced by ZN,β up to exponentially small corrections when N is large (see [] for results in this
direction, and we give a proof for completeness in § 3.1 below). The latter can be decomposed as a
sum over all possible ways of sharing the λ’s between the segments Ah , namely:
V,A
ZN,β
“
N!
V ;A
ZN,N
{N,β ,
N
!
h“0 h
ÿ
śg
0ďN1 ,...Ng ďN
(1.24)
řg
where we have denoted N0 “ N ´ h“1 Nh the number of λ’s put in the segment A0 . So, we can use
our results for the model with fixed filling fractions to analyze the asymptotic behavior of each term
in the sum, and then find the asymptotic expansion of the sum taking into account the interference
of all contributions.
In order to state the result, we need to introduce the Siegel Theta function with characteristics
µ, ν P Cg . If τ be a g ˆ g matrix of complex numbers such that Im τ ą 0, it is the entire function of
v P Cg defined by the converging series:
„ 
´
¯
ÿ
µ
ϑ
pv|τ q “
exp iπpm ` µq ¨ τ ¨ pm ` µq ` 2iπpv ` νq ¨ pm ` µq .
(1.25)
ν
g
mPZ
Among its essential properties, we mention:
‚ for any characteristics µ, ν, it satisfies the diffusion-like equation 4iπBτh,h1 ϑ “ Bvh Bvh1 ϑ.
‚ it is a quasiperiodic function on the lattice Zg ‘ τ pZg q: for any m0 , n0 P Zg ,
„ 
„ 
`
˘ µ
µ
ϑ
pv ` m0 ` τ ¨ n0 |τ q “ exp 2iπm0 ¨ µ ´ 2iπn0 ¨ pv ` νq ´ iπn0 ¨ τ ¨ n0 ϑ
pv|τ q. (1.26)
ν
ν
‚ it has a nice transformation law under τ Ñ pAτ ` BqpCτ ` Dq´1 where A, B, C, D are the
g ˆ g blocks of a 2g ˆ 2g symplectic matrix [Mum84].
‚ when τ is the matrix of periods of a genus g Riemann surface, it satisfies the Fay identity [Fay70].
We define the operator ∇v acting on the variable v of this function. For instance, the diffusion
equation takes the form 4iπBτ ϑ “ ∇b2
v ϑ.
Theorem 1.5 Assume Hypotheses 1.2 and 1.4. Let ‹ “ pµeq rSh sq1ďhďg . Given the coefficients of
tku
the expansion in the fixed filling fraction model from Theorem 1.4, we denote pF‹,β qp`q their tensor of
6
`-th order derivatives with respect to , evaluated at ‹ . Then, the partition function has an asymptotic
expansion of the form:
#
+
„

´ÿ
“
‰¯ ´N ‹
V ;A
V ;A
´k tku ∇v
´8
ZN,β “ ZN,‹ ,β
N T‹,β
ϑ
pv‹,β |τ‹,β q ` OpN q .
(1.27)
0
2iπ
kě0
t0u
In this expression, if X is a vector with g components, T,β rXs “ 1, and for k ě 1:
tku
T,β rXs “
k
ÿ
1
r!
r“1
´â
r pF tmi u qp`i q ¯
ÿ
,β
`i !
i“1
`1 ,...,`r ě1
m1 ,...,mr ě´2
ř
r
i“1 `i `mi “k
řr
¨ X bp
i“1
`i q
,
(1.28)
where ¨ denotes the contraction of tensors. We have also introduced:
t´1u
v‹,β “
pF‹,β q1
2iπ
t´2u
,
τ‹,β “
pF‹,β q2
2iπ
.
(1.29)
Being more explicit but less compact, we may rewrite:
tku “ ∇v ‰
ϑ
T‹,β
2iπ
„

k
ÿ
1
´N ‹
pv‹,β |τ q “
0
r!
r“1
ÿ
´â
r pF tmi u qp`i q ¯
,β
`1 ,...,`r ě1
m1 ,...,mr ě´2
ř
r
i“1 `i `mi “k
´ ÿ
ˆ
`i !
i“1
řr
pm ´ N ‹ qbp
i“1
(1.30)
`i q iπpm´N ‹ q¨τ‹,β ¨pm´N ‹ q`2iπv‹,β ¨pm´N ‹ q
e
¯
.
mPZg
For β “ 2, this result has been derived heuristically to leading order in [BDE00], and to all orders
in [Eyn09], and the arguments there can be extended straightforwardly to all values of β, see e.g.
[Bor11]. Our work justifies their heuristic argument. We exploit the Schwinger-Dyson equations for
the β ensemble with fixed filling fractions taking advantage of a rough control on the large N behavior
of the correlators. The result of Theorem 1.5 has been derived up to op1q by Shcherbina [Shc12] for realś
analytic potentials, with different techniques, based on the representation of 1ďhăh1 ďg |λh,i ´ λh1 ,j |β ,
which is the exponential of a quadratic statistic, as expectation value of a linear statistics coupled
to a Brownian motion. The rough a priori controls on the correlators do not allow at present the
description of the op1q by such methods. The results in [Shc12] were also written in a different form:
F t0u was identified with a combination of Fredholm determinants (see also the physics paper [WZ06]),
whereas this representation does not come naturally in our approach). Also, the step of the analysis
of Section 8 consisting in replacing the sum over nonnegative integers such that N0 ` . . . ` Ng “ N in
(1.24), by a sum over N P Zg , thus reconstructing the theta function, was not performed in [Shc12]
Let us make a few remarks. The 2iπ appears because we used the standard definition of the Siegel
theta function, and should not hide the fact that all terms in (1.30) are real-valued. Here, the matrix:
t´2u
τ‹,β “
HessianpF‹,β q
2iπ
(1.31)
involved in the theta function has purely imaginary entries, and Im τ‹,β is definite positive according
to Theorem 1.4, hence the theta function in the right-hand side makes sense. Notice also that for
it is Zg -periodic in its characteristics µ, hence we can replace ´N ‹ by ´N ‹ ` tN ‹ u, and this is
responsible for modulations of frequency Op1{N q in the asymptotic expansion, and thus breakdown of
the 1{N expansion. Still, ”subsequential” asymptotic expansions in 1{N may occur. For instance, in
7
a symmetric two cuts (g “ 1) model, we have ‹ “ 1{2 and thus the right-hand side is an asymptotic
expansion in powers of 1{N depending on the parity of N .
Let us give the two first orders of (1.30):
t1u
T‹,β rXs “
1 t´2u 3
1 t´1u
t0u
pF‹,β q ¨ X b3 ` pF‹,β q2 ¨ X b2 ` pF‹,β q1 ¨ X,
6
2
(1.32)
and:
t2u
T‹,β rXs
“
1 “ t´2u 3 ‰b2
1 “ t´2u 3
t´1u ‰
pF‹,β q
¨ X b6 `
pF‹,β q b pF‹,β q2 ¨ X b5
72
12
´1 “
¯
1 “ t´1u 2 ‰b2
1
t´2u 3
t0u 1 ‰
t´2u
`
`
pF‹,β q b pF‹,β q `
pF‹,β q
pF‹,β qp4q ¨ X b4
6
8
24
´1 “
1 t´1u 3 ¯
t´1u 2
t0u 1 ‰
b3
pF‹,β q b pF‹,β q ` pF‹,β q ¨ X
`
2
6
´1 “
1 t0u 2 ¯
t0u 1 ‰b2
t1u
pF‹,β q
` pF‹,β q ¨ X b2 ` pF‹,β q1 ¨ X.
`
2
2
(1.33)
If the potential V is independent of β, we observe that ‹ does not depend on β, and it is well-known
[CE06] that the coefficients in the expansion (1.22) have a simple dependence in β:
tku
F,β
tk{2u`1
ÿ ˆ
“
G“0
β
2
˙1´G ´
1´
2 ¯k`2´2G rG,k`2´2Gs
F
.
β
In particular, those coefficients vanish for odd k. The first few ones are:
´β
¯
´β
¯
β
2
t´2u
t´1u
t0u
F,β “ Fr0,0s ,
F,β “
´ 1 Fr0,1s ,
F,β “ Fr1,0s `
` ´ 2 Fr0,2s ,
2
2
2
β
(1.34)
(1.35)
and have been first identified in [WZ06]. In particular, for the argument of the theta function:
v‹,β “
´β
2
´1
¯ pF r0,1s q1
‹
2iπ
r0,0s
,
τ‹,β “
β pF‹ q2
.
2
2iπ
(1.36)
Similarly for the correlators in the fixed filling fraction model, the dependence in β takes the form:
tku
Wn,
px1 , . . . , xn q “
ˆ
tpk´n`2q{2u
ÿ
G“0
β
2
˙1´G´n ´
rG,Ks
1´
2 ¯k`2´2G´n rG;k`2´2G´ns
Wn,
px1 , . . . , xn q.
β
(1.37)
rG,Ks
All coefficients F
and functions Wn, px1 , . . . , xn q can be computed with the β deformation of
the topological recursion formulated by Chekhov and Eynard [CE06], applied to the spectral curve
t0u
determined by the equilibrium measure µeq, and W2, , which encodes the covariance of linear statistics
at leading order in the model with fixed filling fractions. We stress now a point of this theory relevant in
the present case. When V is a polynomial and is close enough to ‹ , the density of the equilibrium
measure can be analytically continued to a hyperelliptic curve of genus g, denoted C and called
spectral curve. Its equation is:
y2 “
g
ź
`
´
´ ρh
` ρh
px ´ α,h
q px ´ α,h
q .
(1.38)
h“0
´
`
Let Ah be the cycle in C surrounding A,h “ rα,h
, α,h
s. The family A “ pAh q1ďhďg can be completed
rG,Ks
by a family of cycles B so that pA, Bq is a symplectic basis of homology of C . The correlators Wn,
are meromorphic functions on Cn , computed recursively by a residue formula on C . In particular, the
analytic continuation of
r0,0s
ω20 px1 , x2 q “ W2, px1 , x2 qdx1 dx2 `
8
2 dx1 dx2
β px1 ´ x2 q2
(1.39)
is the unique 2-form on C , which has vanishing A periods, and has for only singularity a double
pole with leading coefficient β2 and without residue at coinciding points. Then, it is a property of
rG,Ks
the topological recursion that the derivatives of F
can be computed as B-cycle integrals of the
correlators: for any pG, Kq ‰ p0, 0q, p0, 1q,
˛
˛
rG,Ks
rG,Ks p`q
pF
q “
dx1 ¨ ¨ ¨
dx` W`, px1 , . . . , x` q,
(1.40)
B
B
and for any pn, G, Kq:
˛
pWnrG,Ks qp`q px1 , . . . , xn q “
˛
rG,Ks
dxn`1 ¨ ¨ ¨
B
B
dxn`` Wn``, px1 , . . . , xn`` q.
(1.41)
In particular:
r0,0s
pW1, q1 pxqdx “ 2iπ $pxq
(1.42)
where $ is the basis of holomorphic 1-forms on C dual to A, i.e. characterized by
¸
Ah
$h1 “ δh,h1 .
tku
Tβ rXs
This formula at “ ‹ can be used to compute the functions
appearing in (1.28). The
derivation wrt is not a natural operation in the initial model when N is finite, since N h are forced
to be integers in (1.17). We rather show that the coefficients of expansion themselves are smooth
functions of , and thus B makes sense.
t2k`1u
For β “ 2, we remark from (1.34) that the coefficients F‹,β“2 all vanishes, so that we retrieve
the celebrated 1{N 2 expansion in the one-cut regime or in the fixed filling fraction model. This is
in general not true anymore in the multi-cut regime. For instance, we have a term of order 1{N
involving:
1 t´2u
t0u
t1u
(1.43)
T‹,β“2 rXs “ pF‹,β q3 ¨ X b3 ` pF‹,β q1 ¨ X.
6
In a two-cut regime (g “ 1), a sufficient condition for all terms of order N ´p2k`1q to vanish is that
V ;A
V ;A
‹ “ 1{2 and ZN,
“ ZN,1´
, i.e. the potential has two symmetric wells. In this case, we have an
2
expansion in powers of 1{N for the partition function, whose coefficients depends on the parity of
N . In general, we also observe that v‹,β“2 “ 0, i.e. Thetanullwerten appear in the expansion.
1.6
Asymptotic expansion of kernels and correlators
Once the result on large N expansion of the partition function is obtained, we can easily infer the
asymptotic expansion of the correlators and the kernels by perturbing the potential by terms of order
1{N , maybe complex-valued, as allowed by Hypothesis 1.4.
1.6.1
Leading behavior of the correlators
Although we could write down the expansion for the correlators as a corollary of Theorem 1.5, we
bound ourselves to point out their leading behavior. Whereas Wn behaves as OpN 2´n q in the one-cut
regime or in the model with fixed filling fractions, Wn for n ě 3 does not decay when N is large in a
pg ` 1q-cut regime with g ě 1. More precisely:
Theorem 1.6 Assume Hypothesis 1.2, 1.4 and number of cuts pg ` 1q ě 2. We have, for uniform
n
p
convergence when x1 , . . . , xn belongs to any compact of pCzAq
:
„

´ $px q $px q ¯
ˇ
˘
´N ‹ `
1
2
t0u
W2 px1 , x2 q „ W2,‹ px1 , x2 q `
b
¨ ∇b2
ln
ϑ
v‹,β ˇτ‹,β ,
(1.44)
v
0
N Ñ8
dx1
dx2
9
and for any n ě 3:
„

´â
n
ˇ
˘
$pxi q ¯
´N ‹ `
bn
Wn px1 , . . . , xn q „
¨ ∇v ln ϑ
v‹,β ˇτ‹,β .
0
N Ñ8
i“1 dxi
(1.45)
Integrating this result over A-cycles provide the leading order behavior of n-th order moments of the
filling fractions N . We will also describe in Section 8.2 the fluctuations of the filling fractions: we
find that they converge to a discrete Gaussian random variable.
1.6.2
Kernels
We explain in § 6.4 that the following result concerning the kernel – defined in (1.8) – is a consequence
of Theorem 1.3:
Corollary 1.7 Assume Hypothesis 1.2 and 1.4. There exists t ą 0 such that, for any P Eg such
that | ´ ‹ | ă t, the n-kernels in the model with fixed filling fractions have an asymptotic expansion
of the form:
Kn,c, px1 , . . . , xn q “ exp
” ÿ
¯
ı
´ k`2
ÿ 1
tku
´8
Lbn
rW
s
`
OpN
q
,
k! x,c n,
n“1
N ´k
kě´1
where Lx,c is the linear form :
n
ÿ
Lx,c “
ˆ
(1.46)
xj
cj
.
(1.47)
8
j“1
p
Up to a given OpN ´K q, this expansion is uniform for x1 , . . . , xn in any compact of CzA.
´
p , we set Lγ “ , and Lbn is given by:
Hereafter, if γ is a smooth path in CzS
γ
γ
ˆ
tku
Lbn
γ rWn s “
ˆ
dxn Wntku px1 , . . . , xn q.
dx1 ¨ ¨ ¨
γ
γ
A priori, the integrals in the right-hand side of (1.46) depend on the homology class in CzA of paths
8 Ñ xi . A basis of homology cycles in CzA is given by A “ pAh q0ďhďg . We also denote for
convenience “ ph q0ďhďg . We deduce from (1.18) that:
˛
dξ
tku
Wn,
pξ, x2 , . . . , xn q “ δn,1 δk,´1 .
(1.48)
2iπ
A
´
t´1u
Therefore, the only multivaluedness of the right-hand side comes from the first term N dξ W1, pξq,
and given (1.48) and observing that N h are integers, we see that it exactly reproduces the monodromies of the kernels depending on cj .
We now come to the multi-cut regime of the initial model. If X is a vector with g components,
p , let us define:
and L is a linear form on the space of holomorphic functions on CzS
tku
T̃,β rL, Xs “
k
ÿ
1
r!
r“1
ÿ
`1 ,...,`r ě1
m1 ,...,mr ě´2
řr n1 ,...,nr ě0
i“1 `i `mi `ni “k
tku
tm u
´â
r
Lbni rpW i qp`i q s ¯
ni ,
i“1
tku
ni ! `i !
¨ X bp
řr
i“1
`i q
where we took as convention Wn“0, “ F . Then, as a consequence of Theorem 1.5:
10
,
(1.49)
Corollary 1.8 Assume Hypothesis 1.2 and 1.4. With the notations of Corollary 1.7, the n-kernels
have an asymptotic expansion1 :
“ Kn,c,‹ pxqp1 ` OpN ´8 qq
„

´ř
ˇ
˘
‰¯ ´N ‹ `
“
∇v
´k tku
v‹,β ` Lx,c r$sˇτ‹,β
T̃‹,β Lx,c , 2iπ ϑ
kě0 N
0
„

,
ˆ
´ř
ˇ
“ ‰¯ ´N ‹ `
˘
´k T tku ∇v
ˇ
N
ϑ
v
τ
‹,β ‹,β
kě0
‹,β 2iπ
0
´
řn
x
“ j“1 cj 8j and $ is the basis of holomorphic 1-forms.
Kn,c pxq
where Lx,c
(1.50)
A diagrammatic representation for the terms of such expansion was proposed in [BE12, Appendix A].
2
Application to orthogonal polynomials and integrable systems
Since orthogonal polynomials are related to the 1-hermitian matrix model (i.e. β “ 2), our results can
be used to establish the all-order asymptotics of orthogonal polynomials outside the bulk (Theorem 2.2
below). We will illustrate it for orthogonal polynomials with respect to an analytic weight defined on
the whole real line, but it could be applied equally well to orthogonal polynomials with respect to an
analytic weight on a finite union of segments of the real axis.
The leading order asymptotic of orthogonal polynomials is well-known since the work of Deift et
al. [DKM` 97, DKM` 99b, DKM` 99a], using the asymptotic analysis of Riemann-Hilbert problem
which was pioneered in [DZ95]. In principle, it is possible to push the Riemann-Hilbert analysis
beyond leading order, but this approach being very cumbersome, it has not been performed yet to
our knowledge. Notwithstanding, the all-order expansion has a nice structure, and was heuristically
derived by Eynard [Eyn06] based on the general works [BDE00, Eyn09]. In this article, we provide a
proof of those heuristics.
Unlike the Riemann-Hilbert technique which becomes cumbersome to study the asymptotics of
skew-orthogonal polynomials (i.e. β “ 1 and 4) and thus has not been performed up to now, our
method could be applied without difficulty to those values of β, and would allow to justify the heuristics
of Eynard [Eyn01] formulated for the leading order, and describe all subleading orders. In other words,
it provides a purely probabilistic approach to address asymptotic problems in integrable systems. It
also suggest that the appearance of theta functions is not intrinsically related to integrability. In
particular, we see in Theorem 2.2 that for β “ 2, the theta function appearing in the leading order is
associated to the matrix of periods of the hyperelliptic curve C‹ defined by the equilibrium measure.
Actually the theta function is just the basic block to construct analytic functions on this curve, and
this is the reason why it pops up in the Riemann-Hilbert analysis. However, for β ‰ 2, the theta
function comes is associated to pβ{2q times the matrix of periods of C‹ , which might be or not the
matrix of period of a curve, and anyway is not that of C‹ . So, the monodromy problem solved by this
theta function is not directly related to the equilibrium measure, which makes for instance for β “ 1
or 4 its construction via Riemann-Hilbert techniques a priori more involved.
Contrarily to Riemann-Hilbert techniques however, we are not yet in position within our method
to consider the asymptotic in the bulk, at the edges, or the double-scaling limit for varying weights
close to a critical point, or the case of complex-values weights which has been studied in [BM09]. We
hope those technical restrictions to be removable in a near future.
1 We warn the reader that
correlators.
1
denotes a derivative with respect to filling fractions, not with respect to variables of the
11
2.1
Setting
We first review the standard relations between orthogonal polynomials on the real line, random matrices and integrable systems, see e.g. [CG12, Section 5]. In this section, β “ 2 and we omit to
řd
precise it in the notations. Let Vt pλq “ V pλq ` k“1 tk λk . Let pPn,N pxqqně0 be the monic orthogonal
polynomials associated to the weight dwpxq “ dx e´N Vt pxq on B “ R. We choose V and restrict in
consequence tk so that the weight decreases quickly at ˘8. If we denote hn,N the L2 pdwq norm of
a
Pn,N , the polynomials P̂n,N “ Pn,N { hn,N are orthonormal. They satisfy a three-term recurrence
relation:
a
a
(2.1)
xP̂n,N pxq “ hn,N P̂n`1 pxq ` βn,N P̂n pxq ` hn´1,N P̂n´1 pxq.
The recurrence coefficients are solutions of a Toda chain: if we set
un,N “ ln hn,N ,
vn,N “ ´βn,N ,
(2.2)
we have:
Bt1 vn,N “ eun`1,N ´ eun,N ,
Bt1 un,N “ vn,N ´ vn´1,N ,
(2.3)
and the coefficients tk generate higher Toda flows. The recurrence coefficients also satisfy the string
equations:
a
n
hn,N rV 1 pQN qsn,n´1 “ ,
rV 1 pQN qsn,n “ 0,
(2.4)
N
where QN is the semi-infinite matrix:
˛
¨ a
h1,N aβ1,N
‹
˚ β1,N
h2,N aβ2,N
‹
˚
‹
˚
β2,N
h3,N β3,N
(2.5)
QN “ ˚
‹.
‹
˚
..
..
..
‚
˝
.
.
.
Eqns 2.4 determines in terms of V the initial condition for the system (2.3). The partition function
Vt ;R
is the Tau function associated to the solution pun,N ptq, vn,N ptqqně1 of (2.3). The
T ptq “ ZN
partition function itself can be computed as [Meh04, PS11]:
V ;R
ZN
“ N!
N
´1
ź
hj,N .
(2.6)
j“1
We insist on the dependence on N and V by writing hj,N “ hj pN V q. Therefore, the norms can be
retrieved as:
śn
hn pN V q “
j“1 hj pN V q
śn´1
j“1 hj pN V q
N V {pn`1q;R
1 Zn`1
“
n ` 1 ZnN V {n;R
V
sp1`1{nq
1 Zn`1
“
n ` 1 ZnV {s;R
;R
,
s“
n
.
N
(2.7)
The regime where n, N Ñ 8 but s “ n{N remains fixed and positive correspond to the small
dispersion regime in the Toda chain, where 1{n plays the role of the dispersion parameter.
2.2
Small dispersion asymptotics of hn,N
When Vt0 {s0 satisfies Hypotheses 1.2 and 1.3 for a given set of times ps0 , t0 q, Vt {s satisfies the same
assumptions at least for ps, tq in some neighborhood U of ps0 , t0 q, and Theorem 1.5 determines the
Vt ,R
asymptotic expansion of TN ptq “ ZN
up to OpN ´8 q. Besides, we can apply Theorem 1.5 to study
the ratio in the right-hand side of (2.7) when n Ñ 8.
12
Theorem 2.1 In the regime n, N Ñ 8, s “ n{N ą 0 fixed, and Hypotheses 1.2 and 1.3 are satisfied
with soft edges, we have the asymptotic expansion:
un,N
`
1
r0s ˘
r0s
r0s
“ n 2F‹r0s ´ LVt {s rW1,‹ s ` F‹r0s ´ LVt {s rW1,‹ s ` Lb2
Vt {s rW2,‹ s
2

¨ „
ˇ ˘˛
´pn ` 1q ‹ `
ˇτ‹
ϑ
L
r$s
Vt {s
˚
‹
0
˚
‹
„

` ln ˝
‚
´n ‹ ˇˇ ˘
ϑ
p0 τ‹
0
´
ÿ
1¯
rGs
`
pn ` 1q2´2G´m Lbm
´ ln 1 `
Vt {s rWm,‹ s
n
Gě0, mě0
2´2G´mă0
„

ˇ ˘˛
‰¯ ´pn ` 1q ‹ `
“
∇
´k tku
ˇτ‹
ϑ
L
r$s
pn
`
1q
T̃
L
;
‹
Vt {s
Vt {s 2iπ
kě1
˚
‹
0
˚
‹
„

` ln ˝1 `
‚
ˇ ˘
´pn ` 1q ‹ `
ˇ
ϑ
LVt {s r$s τ‹
0
„

¨
´ř
“ ‰¯ ´n ‹ ` ˇ ˘ ˛
´k tku ∇
0ˇτ‹ ‹
T‹ 2iπ ϑ
kě1 n
˚
0
‹
„

´ ln ˚
˝1 `
‚
´n ‹ ` ˇˇ ˘
ϑ
0 τ‹
0
¨
´ř
(2.8)
V {s
Here, ‹ are the filling fractions of µeqt
and LVt {s is the linear form defined by:
˛
LVt {s rf s “
S
dξ Vt pξq
f pξq
2iπ s
(2.9)
We have not performed the expansion of 1{pn ` 1q in powers of 1{n to make the structure more
rGs
transparent. We recall that all the quantities Wm,‹ can be computed from the equilibrium measure
associated to the potential Vt , so making those asymptotic explicit just requires to solve the scalar
t
Riemann-Hilbert problem for µsV
eq . Notice that the number g ` 1 of cuts a priori depends on ps0 , t0 q,
and we do not address the issue of transitions between regimes with different number of cuts (because
we cannot relax at present our off-criticality assumption), which are expected to be universal [Dub08].
We also collect here in one place and for β “ 2 other notations appearing throughout the text:
r0s
rGs
,
W0,‹ “ F‹rGs “ Ft2G´2u
‹
rGs
t2G´2`nu
Wn,‹
“ Wn,
,
‹
τ‹ “
pF‹ q2
,
2iπ
(2.10)
and
T‹tku rXs
T̃‹tku rL ; Xs
“
“
k
ÿ
1
r!
r“1
k
ÿ
1
r!
r“1
rG s
´â
r
pF i qp`i q ¯
ÿ
‹
`1 ,...,`r ě1
G1 ,...,Gr ě0
řr `i `2Gi ´2ą0
i“1 p`i `2Gi ´2q“k
`i !
i“1
řr
¨ X bp
i“1
rG s
´â
r
Lbni rpW i qp`i q s ¯
ÿ
ni ,‹
`1 ,...,`r ě1
G1 ,...,Gr ě0
n1 ,...,nr ě0
řr `i `2Gi ´2`ni ą0
i“1 p`i `2Gi ´2`ni q“k
13
i“1
ni ! `i !
`i q
,
(2.11)
řr
¨ X bp
i“1
`i q
.
(2.12)
2.3
Asymptotic expansion of orthogonal polynomials away from the bulk
The orthogonal polynomials can be computed thanks to Heine formula [Sze39]:
Pn pxq “ µnVt {s;R
n
“ź
‰
px ´ λi q “ K1,1 pxq.
(2.13)
i“1
Hence, as a corollary of Theorem 1.8:
Theorem 2.2 In the regime n, N Ñ 8, s “ n{N ą 0 fixed, and Hypotheses 1.2 and 1.3 are satisfied,
for x P CzS, we have the asymptotic expansion:
Pn pxq
where Lx “
´x
8
rGs ¯
`
˘
Lbm
x rWm,‹ s
1 ` Opn´8 q
m!
mě1 Gě0
„

¯
´ř
ˇ ˘
‰
“
´n ‹ `
∇v
´k tku
Lx r$sˇτ‹
T̃
Lx ; 2iπ ϑ
kě0 n
0
„

,
ˆ
´ř
“ ∇ ‰¯ ´n ‹ ` ˇ ˘
v
´k
tku
ˇ
T
ϑ
0 τ‹
kě0 n
2iπ
0
“ exp
´ ÿ ÿ
n2´2G´m
(2.14)
. Up to a given OpN ´K q, this expansion is uniform for x in any compact of CzS.
We remark that Lx r$s is the Abel map evaluated between the points x and 8.
As such, the results presented in this article do not allow the study of the asymptotic expansion
of orthogonal polynomials in the bulk, i.e. for x P S. Indeed, this requires to perturb the potential
V pλq by a term ´ n1 lnpλ ´ xq having a singularity at x P S, a case going beyond our Hypothesis 1.4.
Similarly, we cannot address at present the regime of transitions between a g cut regime and and
g 1 -cut regime with g ‰ g 1 , because offcriticality was a key assumption in our derivation. Although it
is the most interesting in regard of universality, the question of deriving uniform asymptotics, even at
the leading order, valid for the crossover around a critical point is still open from the point of view of
our methods.
3
3.1
Large deviations and concentration of measure
Restriction to a vicinity of the support
Our first step is to show that the interval of integration in (1.1) can be restricted to a vicinity of
the support of the equilibrium measure, up to exponentially small corrections when N is large. The
proofs are very similar to the one-cut case [BG11], and we remind briefly their idea in § 3.2. Let V
be a regular and confining potential, and µVeq;B the equilibrium measure determined by Theorem 1.1
or Theorem 1.2. We denote S its (compact) support. We define the effective potential by:
ˆ
U V ;B pxq “ V pxq ´ 2 dµVeq;B pξq ln |x ´ ξ|,
Ũ V ;B pxq “ U V ;B pxq ´ inf U V ;B pξq
(3.1)
ξPB
B
when x P B, and `8 otherwise.
Lemma 3.1 If V is regular and confining, we have large deviation estimates: for any F Ď BzS closed
and O Ď BzS open,
1
β
;B
ln µVN,β
rDi λi P Fs ď ´ inf Ũ V ;B pxq,
N
2 xPF
1
β
;B
rDi λi P Os ě ´ inf Ũ V ;B pxq.
lim inf
ln µVN,β
N Ñ8 N
2 xPO
lim sup
(3.2)
N Ñ8
14
(3.3)
We say that V satisfies a control of large deviations on B if Ũ V,B is positive on BzS. Note that Ũ V,B
vanishes at the boundary of S. According to Lemma 3.1, such a property implies that large deviations
outside S are exponentially small when N is large.
Corollary 3.2 Let V be regular, confining, satisfying a control of large deviations on B, and assume
BB X S “ H. Let A Ď B be a finite union of segments such that S Ă Å. There exists ηpAq ą 0 so that:
V ;B
V ;A
ZN,β
“ ZN,β
p1 ` Ope´N ηpAq qq,
(3.4)
and for any n ě 1, there exists a universal constant γn ą 0 so that, for any x1 , . . . , xn P pCzBqn :
´N ηpAq
ˇ V ;B
ˇ
ˇWn px1 , . . . , xn q ´ WnV ;A px1 , . . . , xn qˇ ď śγnn e
.
i“1 dpxi , Bq
(3.5)
It is useful to have a local version of this result:
Corollary 3.3 Let V be regular, confining, satisfying a control of large deviations on B, and assume
BB X S “ H. Let A Ď B be a finite union of segments such that S Ď Å. If a0 is the left (resp. right)
edge of a connected component of A, let us define Aa “ A Y ra, a0 s. For any ε ą 0 small enough, there
exists ηε ą 0 so that, for N large enough and any a Psa0 ´ ε, a0 ` εr, we have:
ˇ
ˇ
ˇ
V ;A ˇ
(3.6)
ˇBa ln ZN a ˇ ď e´N ηε ,
and, for N large enough and any n ě 1 and x1 , . . . , xn P pCzAa q:
ˇ
ˇ
γn e´N ηε
ˇ 1
ˇ
.
ˇBa´ WnV ;Aa px1 , . . . , xn qˇ ď śn
i“1 dpxi , Aa q
(3.7)
From now on, even though we want initially to study the model on BN , we are going first to study
the model on AN , where A is small (but fixed) enlargement of S as allowed above, in particular we
choose A bounded.
Proposition 3.4 For any fixed P E g , the same results holds for the partition function and the
correlators in the fixed filling fraction model.
3.2
Sketch of the proof of Lemma 3.1
We only sketch the proof, since it is similar to [BG11].
řN
Recall that LN “ N ´1 i“1 δλi denotes the normalized empirical measure, either in the initial
model, or in the fixed filling fraction model. We represent:
;B “
µVN,β
Di
λi P Fs “ N
;B
ΥVN,β
pFq
;B
ΥVN,β
pBq
where, for any measurable set X:
„ˆ
ˆ
! Nβ
¯)
NV
V ;B
N ´1 ;B
dξ exp ´
V pξq ` pN ´ 1qβ dLN ´1 pλq ln |ξ ´ λ|
ΥN,β pXq “ µN ´1,β
2
X
B
(3.8)
(3.9)
;B
We first prove a lower bound for ΥVN,β
pXq assuming X contains at least an open interval, of size larger
than some ε ą 0. Let κ1 pV q be the Lipschitz constant for V on B, and:
Xε “ tx P B,
inf |x ´ ξ| ą ε{2u
ξPBzX
15
(3.10)
Using twice Jensen’s inequality, we get
«ˆ
;B
ΥVN,β
pXq
ě
ě
ff
ˆ
¯)
Nβ
sup µN ´1,β
dξ exp ´
V pξq ` pN ´ 1qβ dLN ´1 pηq ln |ξ ´ η|
2
xPXε
x´ε{4
B
«ˆ
ff
ˆ
`
˘
!
)
x`ε{4
NV
κ pV q
´ N2β V pxq` 1 2 ε
N ´1 ;B
dξ exp pN ´ 1qβ
sup e
µN ´1,β
dLN ´1 pλq ln |ξ ´ λ|
x`ε{4
NV
N ´1 ;B
!
xPXε
ě
ε
sup e
2 xPXε
x´ε{4
´ N2β
`
V
κ pV q
pxq` 1 2 ε
˘
B
”ˆ
ı)
!
NV
N ´1 ;B
dLN ´1 pλq Hx,ε pλq
exp pN ´ 1qβ µN
´1,β
(3.11)
B
where we have set:
ˆ
x`ε{4
Hx,ε pλq “
x´ε{4
dξ
ln |ξ ´ λ|
ε{2
(3.12)
For any fixed ε ą 0, Hx,ε is bounded continuous on any compact, so we have by Theorem 1.1 in the
initial model (or Theorem 1.2 in the fixed filling fraction model):
ˆ
`
˘
!
)
κ1 pV q
Nβ
ε
;B
ΥVN,β
pXq ě sup e´ 2 V pxq` 2 ε exp pN ´ 1qβ dµVeq;B pλq Hx,ε pλq ` N Rpε, N q
(3.13)
2 xPXε
B
with limN Ñ8 Rpε, N q “ 0. Letting N Ñ 8, we deduce:
ˆ
´
¯
β
β
1
V ;B
ln ΥN,β pXq ě ´ κ1 pV q ε ´
infε V pxq ´ 2 dµVeq;B pλq Hx,ε pλq
(3.14)
lim inf
N Ñ8 N
4
2 xPX
´
Interchanging the integration over ξ and x, observing that ξ Ñ dµVeq;B pλq ln |ξ ´ λ| is smooth and
then letting ε Ñ 0 we conclude
lim inf
N Ñ8
1
β
;B
ln ΥVN,β
pXq ě ´ inf U V ;B pxq
N
2 xPX
(3.15)
where we have recognized the effective potential of (3.1). To prove the upper bound, we observe that
for any M ą 0,
„ˆ

ˆ
! Nβ
NV
`
˘)
V ;B
N ´1 ;B
´1
ΥN,β pXq ď µN ´1,β
dξ exp ´
V pξq ´ pN ´ 1qβ dLN ´1 pλq ln max |ξ ´ λ|, M
2
X
B
(3.16)
`
˘
As λ Ñ ln min |ξ ´ λ|, M ´1 is bounded continuous on compacts, we can use Theorem 1.1 in the
initial model (or Theorem 1.2 in the fixed filling fraction model) to deduce that for any ε ą 0
ˆ
ˆ
)
! Nβ
`
˘
2
;B
ΥVN,β
pXq ď
dξ exp ´
V pξq ´ pN ´ 1qβ dµVeq;B pλq ln max |ξ ´ λ|, M ´1 ` N M ε ` eN R̃pε,N q
2
X
B
(3.17)
with
NV
1
N ´1 ;B
V ;B
lim sup R̃pε, N q “ lim sup 2 ln µN
(3.18)
´1,β pdpLN ´1 , µeq q ą εq ă 0.
N Ñ8
N Ñ8 N
`
˘
´
Moreover, ξ Ñ V pξq ´ dµVeq;B pλq ln max |ξ ´ λ|, M ´1 is bounded continuous so that a standard
Laplace method yields
ˆ
!β ´
¯)
1
;B
ln ΥVN,β
pXq ď ´ inf
V pξq ´ dµVeq;B pλq ln |ξ ´ λ| _ M ´1 .
(3.19)
lim inf
N Ñ8 N
ξPX 2
B
`
´
Finally, we conclude by monotone convergence theorem which implies that dµVeq;B pλq ln max |ξ ´
˘
´
λ|, M ´1 increases as M goes to infinity towards dµVeq;B pλq ln |ξ ´ λ|.
16
3.3
Concentration of measure and consequences
We will need rough a priori bounds on the correlators, which can be derived by purely probabilistic
methods. This type of result first appeared in the work of [dMPS95, Joh98] and more recently
[KS10, MMS12]. Given their importance, we find useful to prove independently the bound we need
by elementary means.
Hereafter, we will say that a function f : R Ñ C is b-Lipschitz if
κb pf q “ sup
x‰y
|f pxq ´ f pyq|
ă 8.
|x ´ y|b
(3.20)
Our final goal is to control pLN ´ µeq qrϕs for a class of functions ϕ which is large enough, in particular
contains analytic functions on a neighborhood of the interval of integration A. This problem can be
settled by controlling the ”distance” between LN and µeq for an appropriate notion of distance. We
introduce the pseudo-distance between probability measures:
¨
Dpµ, νq “ ´
drµ ´ νspxqdrµ ´ νspyq ln |x ´ y|
(3.21)
which can be represented in terms of Fourier transform of the measures by:
ˆ 8
ˇ2
ds ˇˇ
µ ´ νpqpsqˇ
pp
Dpµ, νq “
|s|
0
(3.22)
Since LN has atoms, its pseudo-distance to another measure is in general infinite. There are several
methods to circumvent this issue, and one of them, that we borrow from [MMS12], is to define a regr u (see the beginning of § 3.4.1 below) from LN . Then, the result of concentration,
ularized measure L
N
takes the form:
Lemma 3.5 Let V be regular, C 3 , confining, satisfying a control of large deviations on A. There
exists C ą 0 so that, for t small enough and N large enough:
‰
;A “
r u , µV ;A s ě t ď eCN ln N ´N 2 t2 .
µVN,β
DrL
(3.23)
N
eq
We prove it in § 3.4.1 below. The assumption V of class C 3 ensures that the effective potential (3.1)
defined from the equilibrium measure is a 21 -Lipschitz function (and even Lipschitz if all edges are
soft) on the compact set A, as one can observe on (A.4) given in Appendix A.
This lemma allows a priori control of expectation values of test functions:
Corollary 3.6 Let V be regular, C 3 , confining, satisfying a control of large deviations on A. Let
b ą 0, and assume ϕ : R Ñ C is a b-Lipschitz function with constant κb pϕq, and such that:
´ˆ
¯1{2
|ϕ|1{2 :“
ds |s| |ϕpsq|
p 2
ă 8.
(3.24)
R
Then, there exists C3 ą 0 such that, for t small enough and N large enough:
ˇ
”ˇ ˆ
ı
2κb pϕq
ˇ
;A ˇ
C3 N ln N ´N 2 t2
.
µVN,β
`
t
|ϕ|
ˇ drLN ´ µVeq;A spxq ϕpxqˇ ě
1{2 ď e
2b
pb ` 1qN
A
(3.25)
As a special case, we can obtain a rough a priori control on the correlators:
Corollary 3.7 Let V be regular, C 3 , confining and satisfying a control of large deviations on A. Let
D1 ą 0, and:
c
a
| ln δ|
ln N
D1
wN “
,
f pδq “
,
dpx, Aq “ inf |x ´ ξ| ě ?
(3.26)
ξPA
N
δ
N ln N
17
There exists a constant γ1 pA, D1 q ą 0 so that, for N large enough:
ˇ
ˇ
`
˘
ˇW1 pxq ´ N W t´1u pxqˇ ď γ1 pA, D1 q wN f dpx, Aq .
1
(3.27)
Similarly, for any n ě 2, there exists constants γn pA, D1 q ą 0 so that, for N large enough:
n
ź
ˇ
ˇ
`
˘
n
ˇWn px1 , . . . , xn qˇ ď γn pA, D1 q wN
f dpxi , Aq .
(3.28)
i“1
In the pg ` 1q-cut regime with g ě 1, we denote pSh q0ďhďg the connected components of the support
Ťg
`
of µVeq;B , and we take A “ h“0 Ah , where Ah “ ra´
h , ah s Ď B are pairwise disjoint bounded segments
N
such that Sh Ď Åh . For any configuration λ P A , we denote Nh the number of λi ’s in Ah , and
N “ pNh q1ďhďg . The following result gives an estimate for large deviations of N away from N ‹ in
the large N limit.
Corollary 3.8 Let A be as above, and V be C 3 , confining, satisfying a control of large deviations on
A, and leading to a pg ` 1q-cut regime. There exists a positive constant C such that, for N large
enough and uniformly in t:
?
‰
2
;A “
µVN,β
|N ´ N ‹ | ą t N ln N ď eN ln N pC´t q .
(3.29)
As an outcome of this article, we will be more precise in Section 8.2 about large deviations of filling
fractions when the potential satisfies the stronger Hypotheses 1.1-1.4.
3.4
3.4.1
Large deviation of LN : distance (Lemma 3.5)
Regularization of LN
We start by following an idea introduced by Maı̈da and Maurel-Segala [MMS12, Proposition 3.2]. Let
σN , ηN Ñ 0 be two sequences of positive numbers. To any configuration of points λ1 ď . . . ď λN in
r1 , . . . , λ
rN by the formula:
A, we associate another configuration λ
r 1 “ λ1 ,
λ
ri`1 “ λ
ri ` maxpλi`1 ´ λi , σN q,
λ
(3.30)
It has the properties:
@i ‰ j,
ri ´ λ
r j | ě σN ,
|λ
ri ´ λ
rj |,
|λi ´ λj | ď |λ
ri ´ λi | ď pi ´ 1qσN .
|λ
(3.31)
r N “ N ´1 řN δ r the new counting measure. Then, we define L
r u be the convolution
Let us denote L
N
i“1 λi
r N with the uniform measure on r0, ηN σN s.
of L
r u , which has the advantage of
We are going to compare the logarithmic energy of LN to that of L
N
having no atom. We may write by (3.31):
¨
¨
r N pxqdL
r N pyq ln |x ´ y| “ Σ∆ pL
rN q
Σ∆ pLN q “
dLN pxqdLN pyq ln |x ´ y| ď
dL
(3.32)
x‰y
x‰y
and, if we denote U, U 1 are two independent random variables uniformly distributed on r0, 1s, we find:
ˆ
” ´
1 ¯ı
u
r
r
r N pxqdL
r N pyq E ln 1 ` ηN σN U ´ U
Σ∆ pLN q ´ Σ∆ pLN q “
dL
px ´ yq
x‰y
ˆ
η
σ
r N pxqdL
r N pyq N N ď ηN ,
ď
dL
|x ´ y|
x‰y
18
ri ’s enforced in (3.30). Eventually, we compute with Σpµq “
thanks to the minimal distance between λ
˜
ln |x ´ y|dµpxqdµpyq,
¨
r u q ´ ΣpL
ru q “ ´
r u pxqdL
r u pyq ln |x ´ y|
Σ∆ pL
dL
N
N
N
N
x“y
¯
‰
1 ´3
1 “
´ 3 lnpηN σN q .
´ E ln |ηN σN pU ´ U 1 q| “
N
N 2
“
(3.33)
Besides, if b ą 0 and ϕ : A Ñ C is a b-Lipschitz function with constant κb pϕq, we have by (3.31):
N
ˇˆ
ˇ
ÿ
2κb pϕq
ˇ
r u spxq ϕpxqˇˇ ď κb pϕq
pi ´ 1qb rσN p1 ` 2ηN qsb ď
pN σN qb
ˇ drLN ´ L
N
N i“1
p1 ` bq
A
3.4.2
(3.34)
Large deviations of LuN
r u from the equilibrium measure
We would like to estimate the probability of large deviations of L
N
V
;A
µeq “ µVeq;A . We need first a lower bound on ZN,β similar to that of [AG97] obtained by localizing the
V ;A
ordered eigenvalues at a distance N ´3 of the quantiles λcl
i of the equilibrium measure µeq , which are
defined as:
!
`
˘
i )
V ;A
.
(3.35)
λcl
“
inf
x
P
A,
µ
r´8,
xs
ě
i
eq
N
V ;A
is continuous on the interior of its support, and diverge only at hard edges, where
Since V is C 2 , dµeq
it blows at most like the inverse of a squareroot. Therefore, there exists a constant C ą 0 such that,
for N large enough:
C
cl
|λcl
.
(3.36)
i ´ λi´1 | ě
N2
Then, since V is a fortiori C 1 on A compact,
ˆ
V ;A
ZN,β
ź
ě N!
|δi |ďN ´3 1ďiăjďN
cl
β
|λcl
i ´ λ j ` δi ´ δj |
e´
βN
2
V pλcl
i `δi q
dδi
i“1
ź
ě N ! N ´3N e´C1 N
N
ź
cl β
|λcl
i ´ λj |
N
ź
e´
Nβ
2
řN
i“1
V pλcl
i q
,
(3.37)
i“1
1ďiăjďN
for some constant C1 ą 0. Therefore, since:
¨
ln |x ´
ÿˆ
y|dµVeq;A pxqdµVeq;A pyq
iăj
V ;A
ZN,β
ě exp
ˆ
λcl
j`1
ď
xďy
we find:
λcl
i`1
!β ´
λcl
i
λcl
j
ln |x ´ y|dµVeq;A pxqdµVeq;A pyq
ď
1 ÿ
cl
ln |λcl
i ´ λj |
N 2 iăj´1
ď
1 ÿ
1 ´N2 ¯
cl
ln
,
ln |λcl
i ´ λj | `
2
N iăj
N
C
(3.38)
¯)
´ C2 N ln N ´ N 2 ErµVeq;A s .
(3.39)
2
for some positive constant C2 and with the energy introduced in (1.10).
r u , µV ;A s ě tu. We have:
Now, let us denote SN ptq the event tDrL
eq
N
;A
µVN,β
rSN ptqs
“
1
V ;A
ZN,β
ˆ
e
βN 2
2
`˜
x‰y
˘ź
N
´
dLN pxqdLN pyq ln |x´y|´ dLN pxq V pxq
SN ptq
i“1
19
dλi ,
(3.40)
and using the comparisons (3.32)-(3.34), we find, with the notations of Theorem 1.2:
β
;A
µVN,β
rSN ptqs
ď
ď
e 2 RN
ˆ
e
βN 2
2
`˜
˘ N
´
r u pxqdL
r u pyq ln |x´y|´ dL
r u pxq V pxq ź
dL
N
N
N
V ;A
ZN,β
SN ptq
`
˘
β
2
V ;A
ˆ
e 2 RN ´N Erµeq s
V ;A
ZN,β
dλi
i“1
e´
βN 2
2
`
˘ N
´
r u ,µeq s` dL
r u pxq U V ;A pxq ź
DrL
N
N
SN ptq
dλi , (3.41)
i“1
where:
3N
´ 3N lnpηN σN q,
(3.42)
2
and the effective potential U V ;A was defined in (3.1). Since U V ;A is at least 12 -Lipschitz on A (and
even 1-Lipschitz if all edges are soft),we find:
`
˘
1{2
β
;A
ˆ
N
RN `κ1{2 pU V ;A q N 5{2 σN ´N 2 ErµV
ź
2
eq s
2
βN
V ;A
e
r u ,µeq s
;A
´ βN
DrL
N
2
µVN,β
rSN ptqs ď
e
e´ 2 U pλi q dλi .
V ;A
ZN,β
SN ptq
i“1
(3.43)
We now use the lower bound (3.39) for the partition function, and the definition of the event SN ptq,
in order to obtain:
´ˆ
¯N
1{2
βN
β
β
V ;A
2 2
V ;A
pRN `κ1{2 pU q N 5{2 σN `C2 N ln N ´N 2 t2 q
2
dλ e´ 2 U pλq
ď e 2 pR̃N `C2 N ln N ´N t q ,
µN,β rSN ptqs ď e
RN “ N 3 σN κ1 pV q ` N 2 ηN `
A
(3.44)
with:
1{2
R̃N “ RN ` κ1{2 pU q N 5{2 σN `
2N
ln `pAq.
β
(3.45)
Indeed, since U V ;A is nonnegative on A, we observed that the integral in bracket is bounded by the
total length `pAq of the range of integration, which is here finite. We now choose:
σN “
1
,
N3
1
ηN “ ? ,
N
(3.46)
which guarantee that R̃N P OpN ln N q. Thus, there exists a positive constant C3 such that, for N
large enough:
β
2 2
;A
(3.47)
µVN,β
rSN ptqs ď e 2 pC3 N ln N ´N t q ,
which concludes the proof of Proposition 3.5. We may rephrase this result by saying that the probaa
bility of SN ptq becomes small for t larger than wN “ 2C3 ln N {N .
l
3.5
3.5.1
Large deviations for test functions
Proof of Corollary 3.6
Since ϕ is b-Lipschitz, we can use the comparison (3.34) with σN “ N ´3 chosen in (3.46):
ˇˆ
ˇ
ˇ
r u spxq ϕpxqˇˇ ď 2κb pϕq
ˇ drLN ´ L
N
pb ` 1qN 2b
A
Then, we compute:
ˇˆ
ˇ
ˇ
r u ´ µeq spxq ϕpxqˇˇ “
ˇ drL
N
A
ď
ˇˆ
ˇ
` pu
˘
ˇ
ˇ
r ´ µx
psq
ϕp´sq
p
ˇ ds L
ˇ
eq
N
R
´ ˆ ds ˇ
ˇ2 ¯1{2
p
r u ´ µx
ˇpL
ˇ
qpsq
,
|ϕ|1{2
eq
N
R |s|
20
(3.48)
(3.49)
and we recognize in the last factor the definition (3.22) of the pseudo-distance:
ˇˆ
ˇ ?
ˇ
r u ´ µeq spxq ϕpxqˇˇ ď 2 |ϕ|1{2 DrL
r u , µeq s.
ˇ drL
N
N
(3.50)
A
Corollary 3.6 then follows from this inequality combined with Lemma 3.5.
3.5.2
Bounds on correlators and filling fractions (Proof of Corollary 3.7 and 3.8)
If µ is a probability measure, let Wµ denote its Stieltjes transform. We have:
ˆ
1A pξq
rWLN ´ Wµeq spxq “
drLN ´ µeq spξq ψx pξq,
ψx pξq “ ψxR pξq ` iψxI pξq “
x´ξ
A
(3.51)
Since ψx is 1-Lipschitz with constant κ1 pψx q “ d´2 px, Aq, we have for N large enough:
ˇ
ˇ
ˇrWL ´ W r u spxqˇ ď
N
L
N
1
N2
d2 px, Aq
(3.52)
r u is included in
We now focus on estimating WLr u ´ Wµeq . Since the support of L
N
N
dpx, Aq ď 1{N 2 u,
A1{N 2 “ tx P R,
(3.53)
we have the freedom to replace ψx‚ by any function φ‚x which coincides with ψx‚ on A1{N 2 . We also
find:
ˇ
ˇ ? `
˘
I
ru
ˇrW r u ´ Wµeq spxqˇ ď 2 |φR
(3.54)
x |1{2 ` |φx |1{2 DrLN , µeq s
L
N
Let φ : R Ñ R be C 1 function which decays as φpxq P Op1{x2 q when |x| Ñ 8. We observe:
ˆ
ˆ
1 p1
p 2 ds “
|φ|21{2 “
|s| |φpsq|
|φ psq|2 ds
|s|
R
R
ˆ
ˆ
|ξ1 ´ ξ2 | 1
“ ´2
ln |ξ1 ´ ξ2 | φ1 pξ1 qφ1 pξ2 qdξ1 dξ2 “ ´2
ln
φ pξ1 qφ1 pξ2 qdξ1 dξ2
M
R2
R2
ˆ ˇ ´
ξ1 ´ ξ2 ¯ˇˇ 1
ˇ
(3.55)
ď 2
ˇ |φ pξ1 q| |φ1 pξ2 q|dξ1 dξ2
ˇ ln
M
R2
where M ą 0 can be chosen arbitrarily. Let ax,h P Ah the point such that dpx, Ah q “ |x ´ ax,h |. We
I
claim that, for dpx, Aq small enough, we can always choose φR
x and φx such that:
g
ÿ
ˇ ‚ 1 ˇ
ˇpφx q pξqˇ ď
1
2 ` dpx, A q2
pξ
´
a
q
x,h
h
h“0
(3.56)
This family (indexed by x) of functions is uniformly bounded by pg ` 1q{ξ 2 at 8, which is integrable
at 8. Therefore, we can choose M in (3.55) independent of x so that:
ˆ
´ξ ´ ξ ¯ ˇ
ˇˇ
ˇ
1
2 ˇ ‚ 1
2 ln
|φ‚x |21{2 ď 1 ´
pφx q pξ1 qˇ ˇpφ‚x q1 pξ2 qˇdξ1 dξ2
(3.57)
M
R2
If we plug the bound (3.56) in the right-hand side, the integral can be explicitly computed and we
find a finite constant D ą 0 which depends only on A, such that:
|φ‚x |21{2 ď
D ln dpx, Aq
d2 px, Aq
when x approaches A. Combining (3.52)-(3.54)-(3.58) with Lemma 3.5, we find:
ˇ1
ˇ
ˇ
‰ˇˇ
ˇ
ˇ ;A “
ˇ
t´1u
pxqˇ “ ˇµVN,β
WLN pxq ´ Wµeq pxq ˇ
ˇ W1 pxq ´ W1
N
c
a
| ln dpx, Aq|
1
ln N
ď
` 2D
2
N d px, Aq
N
dpx, Aq
21
(3.58)
(3.59)
If we restrict ourselves to x P CzA such that:
dpx, Aq ě ?
D1
N ln N
(3.60)
for some constant D1 , then:
c
a
ˇ
ˇ1
| ln dpx, Aq|
ln N
ˇ
ˇ
t´1u
1
pxqˇ ď p2D ` D q
.
ˇ W1 pxq ´ W1
N
N
dpx, Aq
(3.61)
Now let us consider the higher correlators. For any n ě 2, WnV ;A is the expectation value of some
‰
;A “
homogeneous polynomial of degree 1 in the quantities pWLN ´Wµeq qpxi q and µVN,β
pWLN ´Wµeq qpxi q .
Accordingly, Lemma 3.5 yields:
a
n
´ ln N ¯n{2 ź
| ln dpxi , Aq|
|Wn px1 , . . . , xn q| ď γn
.
(3.62)
N
dpxi , Aq
i“1
for some constant γn ą 0, which depends only on A. This concludes the proof of Corollary 3.7.
Similarly, to have a hand on filling fraction, we write:
ˆ
Nh ´ N ‹,h “ N drLN ´ µeq spξq 1Ah pξq.
(3.63)
After replacing the function 1Ah by a smooth function which assumes the value 1 on Ah , vanishes on
Ah1 for h1 ‰ h, and has compact support, we can apply Corollary 3.6 to deduce Corollary 3.8.
l
4
Schwinger-Dyson equations for β ensembles
Ťg
Let A “ h“0 Ah be a finite union of pairwise disjoint bounded segments, and V be a C 1 function of
;A
A. Schwinger-Dyson equation for the initial model µVN,β
can be derived by integration by parts. Since
the result is well-known (and has been reproved in [BG11]), we shall give them without proof. They
can be written in several equivalent forms, and here we recast them in a way which is convenient for
our analysis. We actually assume that V extends to a holomorphic function in a neighborhood of A,
so that they can written in terms of contour integrals of correlators, and an extension to V harmonic
will be mentioned in § 6.2. We introduce (arbitrarily for the moment) a partition BA “ pBAq` YpBAq
9
´
of the set of edges of the range of integration, and
Lpxq “
ź
px ´ aq,
L1 px, ξq “
aPpBAq´
Lpxq ´ Lpξq
,
x´ξ
L2 px; ξ1 , ξ2 q “
L1 px, ξ1 q ´ L1 px, ξ2 q
. (4.1)
ξ1 ´ ξ2
Theorem 4.1 Schwinger-Dyson equation at level 1. For any x P CzA, we have:
`
˘2 ´
2¯
W2 px, xq ` W1 pxq ` 1 ´
Bx W1 pxq
β
˛
dξ Lpξq V 1 pξq W1 pξq
2 ÿ Lpaq
V ;A
´N
´
Ba ln ZN,β
x´ξ
β
x´a
A 2iπ Lpxq
aPpBAq`
˛
´
¯
2
dξ L2 px; ξ, ξq
` 1´
W1 pξq
β A 2iπ
Lpxq
‹
˘
dξ1 dξ2 L2 px; ξ1 , ξ2 q `
´
W2 pξ1 , ξ2 q ` W1 pξ1 qW1 pξ2 q “ 0.
2
Lpxq
A2 p2iπq
And similarly, for higher correlators:
22
(4.2)
Theorem 4.2 Schwinger-Dyson equation at level n ě 2. For any x, x2 , . . . , xn P CzA, if we denote
I “ v2, nw, we have:
´
2¯
W|J|`1 px, xJ qWn´|J| px, xIzJ q ` 1 ´
Bx Wn px, xI q
β
JĎI
˛
dξ Lpξq V 1 pξq Wn pξ, xI q
2 ÿ Lpaq
´N
´
Ba Wn´1 pxI q
x´ξ
β
x´a
A 2iπ Lpxq
aPpBAq`
˛
˛
´
2¯
2ÿ
dξ Lpξq Wn´1 pξ, xIztiu q
dξ L2 px; ξ, ξq
`
1
´
`
Wn pξ, xI q
β iPI A 2iπ Lpxq px ´ ξqpxi ´ ξq2
β A 2iπ
Lpxq
¯
ÿ
dξ1 dξ2 L2 px; ξ1 , ξ2 q ´
W
pξ
,
x
qW
pξ
,
x
q
W
pξ
,
ξ
,
x
q
`
1
J
2
n`1
1
2
I
|J|`1
n´|J|
IzJ
p2iπq2
Lpxq
JĎI
Wn`1 px, x, xI q `
‹
´
A2
ÿ
(4.3)
“ 0.
The last line in (4.2) or (4.3) is a rational fraction in x, with poles at a P pBAq` , whose coefficients
are linear combination of moments of λi .
řg
As a matter of fact, if N P r0, N sg so that |N | “ h“1 Nh ď N , the correlators in the model with
;B
fixed filling fractions µVN,“N
{N,β satisfy the same equations. Indeed, in the process of integration by
parts, one does not make use of the information about the location of the λi ’s. By linearity, in a model
V ;A
N where is random, the partition function Z “ ErZN,N
{N,β s and the correlators Wn px1 , . . . , xn q “
;A
ErWN {N,n px1 , . . . , xn qs satisfy the same equations. The initial model µVN,β
and the model with fixed
filling fractions are just special cases of the model with random filling fractions.
p
When g ě 1, we denote A “ pAh q1ďhďg a family of contours surrounding Ah in CzA,
and introduce
the vector-valued linear operator:
˛
¯
´˛
dξ
dξ
f pξq, . . . ,
f pξq
(4.4)
LA rf s “
Ag 2iπ
A1 2iπ
p
on the space of holomorphic functions in CzA.
For any m P v1, nw, We denote:
Wn|m px1 , . . . , xn´m q “ Lbm
A rWn px1 , . . . , xn´m , ‚qs,
(4.5)
which means that we integrate the remaining m variables on A-cycles. By definition, if we denote
peh q1ďhďg the canonical basis of Cg ,
Wn|m px1 , . . . , xn´m q “
ÿ
;A
µVN,‚,β
”
n
ź
i“n´m`1
1ďh1 ,...,hm ďg
Nhi
n´m
ź
j“1
Tr
m
1 ı â
eh .
xj ´ Λ c i“1 i
(4.6)
In particular, Wn|n is the tensor of n-th order cumulants of the numbers Nh of λ’s in the segment
;A
Ah . We take as convention Wn|0 px1 , . . . , xn q “ Wn px1 , . . . , xn q. Here, µVN,‚,β
denotes the probability
measure in a model with N λ’s and random filling fractions. If P Eg and if we specialize to the model
with fixed filling fraction , we have:
Wn|m px1 , . . . , xn´m q “ δn,1 δm,1 N .
(4.7)
In general, we deduce from Theorem 4.2:
Corollary 4.3 For any n ě 1 and m P v0, n ´ 1w, for any x, x2 , . . . , xn´m P CzA, if we denote
23
I “ v2, n ´ mw, we have:
ˆ
Wn`1|m px, x, xI q `
ÿ
JĎI
0ďm1 ďm
˙
m
W|J|`1`m1 |m1 px, xJ q b Wn´|J|´m1 |m´m1 px, xIzJ q
m1
˛
´
2¯
dξ Lpξq V 1 pξq Wn|m pξ, xI q
` 1´
Bx Wn|m px, xI q ´ N
β
x´ξ
A 2iπ Lpxq
˛
ÿ
ÿ
2
Lpaq
2
dξ Lpξq Wn´1|m pξ, xIztiu q
´
Ba Wn´1|m pxI q `
β
x´a
β iPI A 2iπ Lpxq px ´ ξqpxi ´ ξq2
aPpBAq`
˛
‹
´
2¯
dξ L2 px; ξ, ξq
dξ1 dξ2 L2 px; ξ1 , ξ2 q ´
` 1´
Wn|m pξ, xI q ´
Wn`1|m pξ1 , ξ2 , xI q
2
β A 2iπ
Lpxq
Lpxq
A2 p2iπq
ˆ
˙
¯
ÿ
m
1 |m1 pξ1 , xJ q b Wn´|J|´m1 |m´m1 pξ2 , xIzJ q
`
W
“
|J|`1`m
m1
JĎI
(4.8)
0.
0ďm1 ďm
Proof. Straightforward from (4.3), once we notice that the integrals over a closed cycle of a total
derivative or a holomorphic integrand in neighborhoods of A in C give a zero contribution.
l
5
Fixed filling fractions: 1{N expansion of correlators
5.1
Norms on analytic functions and assumptions
In this Section, we analyze the Schwinger-Dyson equations in the model with random filling fractions
(which contains the model with fixed filling fractions as special case) and the following assumptions:
Hypothesis 5.1
`
‚ A is a disjoint finite union of bounded segments Ah “ ra´
h , ah s.
‚ (Real-analyticity) V : A Ñ R extends as a holomorphic function in a neighborhood U Ď C of A.
‚ (1{N expansion for the potential) There exists a sequence pV tku qkě0 of holomorphic functions
in U and constants pv tku qkě0 , so that, for any K ě 0:
K
ˇ
ˇ
ÿ
ˇ
ˇ
sup ˇV pξq ´
N ´k V tku pξqˇ ď v tKu N ´pK`1q .
ξPU
(5.1)
k“0
t´1u
‚ (g ` 1-cut regime) W1
pxq “ limN Ñ8 N ´1 W1 pxq exists, is uniform for x in any compact
of CzA, and extends to a holomorphic function on CzS, where S is a disjoint finite union of
segments Sh “ rαh´ , αh` s Ď Ah .
‚ (Offcriticality) ypxq “
pV t0u q1 pxq
2
t´1u
´ W1
ypxq “ Spxq
pxq takes the form:
g b
ź
`
´
px ´ αh` qρh px ´ αh´ qρh ,
(5.2)
h“0
where S does not vanish on A, αh‚ are all pairwise distinct, and ρ‚h “ 1 if αh‚ P BA, and ρ‚h “ ´1
else.
24
‚ (1{N expansion for correlations of filling fractions) For any n ě 1, there exist a sequence
tku
tku
pWn|n qkěn´2 of elements of pCg qbn , and positive constants pwn|n qkěn´2 , so that, for any K ě
´1, we have:
K
ˇ
ˇ
ÿ
ˇ
tKu
tku ˇ
(5.3)
N ´k Wn|n ˇ ď N ´pK`1q wn|n
ˇWn|n ´
k“n´2
We say that Hypothesis 5.1 holds up to opN ´K q if we only have a 1{N expansion of the potential at
least up to opN ´K q, and an asymptotic expansion for correlations of filling fractions up to opN ´pK´1q q.
Remark 5.2 In the model with fixed filling fractions, this last point is automatically satisfied since
t´1u
Wn|n is given by (4.7). Then, W1
pxq is the Stieltjes transform of the equilibrium measure determined by Theorem 1.2, and Hypothesis 5.1 then constrains the choice of V and .
We fix once for all a neighborhood U of A so that S ´1 p0q X U “ H, and contours A “ pAh q1ďhďg
surrounding Ah in U.
pnq
Definition 5.3 If δ ą 0, we introduce the norm k ¨ kδ on the space Hm1 ,...,mn pAq of holomorphic
i
functions on pCzAqn which behave like Op1{xm
i q when xi Ñ 8:
kf kδ “
sup
|f px1 , . . . , xn q| “
dpx,Aqěδ
sup
|f px1 , . . . , xn q|,
(5.4)
dpxi ,Aq“δ
pnq
pnq
the last equality following from the maximum principle. If n ě 2, we denote Hm “ Hm,...,m .
p1q
From Cauchy residue formula, we have a naive bound on the derivatives of a function f P H1 in
terms of f itself:
2m`1 C
(5.5)
kBxm f pxqkδ ď m`1 kf kδ{2 .
δ
Definition 5.4 We fix once for all a sequence δN of positive numbers, so that:
ln1{3 δN ´ ln N ¯1{3
“0
N Ñ8
δN
N
lim
(5.6)
pnq
If f P Hm pAq is a sequence of functions indexed by N , we will denote f P OpRN pδqq when, for any
ε ą 0, there exists a constant Cpεq ą 0 independent of δ and N , so that:
kf kδ ď Cpεqδ ´ε RN pδq
(5.7)
for N large enough and δ small enough but larger than δN .
This choice will be justified at the end of § 5.3.1, and we can simplify the condition by taking δN of
order N ´1{3`ε for some ε ą 0 arbitrarily small. We notice it also guarantees the assumptions dpx, Aq
not much smaller than pN ln N q´1{2 as it appears in Corollary 3.7.
Definition 5.5 If X is an element of pCg qbn , we define its norm as:
ÿ
|X| “
|Xh1 ,...,hn |.
1ďh1 ,...,hn ďg
25
(5.8)
To perform the asymptotic analysis to all order, we need a rough a priori estimate on the correlators.
We have established (actually under weaker assumptions on the potential) in § 3.3 that:
´
W1 ´
¯
`?
˘
1
t´1u
W1
P O N ln N δ ´θ
N
(5.9)
and for any n ě 2 and m P v0, nw:
`
˘
Wn|m P O pN ln N qn{2 δ ´pn´mqθ
(5.10)
with exponent θ “ 1.
Our goal in this section is to establish under those assumptions Proposition 5.5 below about the
1{N expansion for the correlators. We are going to recast the Schwinger-Dyson equations in a form
which makes the asymptotic analysis easier. We already notice that it is convenient to choose
pBAq˘ “ ta‚h P pBAq,
ρ‚h “ ˘1u,
(5.11)
as bipartition of BA to write down the Schwinger-Dyson equation, since the terms involving Ba ln Z
and Ba Wn´1 for a P pBAq` will be exponentially small according to Corollary 3.3. If a “ a‚h , we denote
αpaq “ αh‚ .
5.2
5.2.1
Some relevant linear operators
The operator K
p1q
We introduce the following linear operator defined on the space H2 pAq:
Kf pxq “
t´1u
2W1
pxqf pxq
1
´
Lpxq
˛
˛
where:
P t´1u px; ξq “
A
A
ı
dξ ” Lpξq pV t0u q1 pξq
` P t´1u px; ξq f pξq,
2iπ
x´ξ
dη
t´1u
2L2 px; ξ, ηq W1
pηq
2iπ
(5.12)
(5.13)
ś
t´1u
We remind that Lpxq “ aPpBAq´ px ´ αpaqq and L2 was defined in (4.1). Notice that W1
pxq „ 1{x
when x Ñ 8, and P t´1u px, ξq is a polynomial in two variables, of maximal total degree |pBAq´ | ´ 2.
Hence:
p1q
p1q
K : H2 pAq Ñ H1 pAq.
(5.14)
Notice also that:
d
˘1
V t0u pxq
L̃pxq
t´1u
´ W1
pxq “ Spxq
,
ypxq “
2
Lpxq
`
where L̃pxq “
ś
(5.15)
´ αpaqq, and by offcriticality assumption the zeroes of S are away from A.
b
Lpxq
px
´
αpaqq
“
L̃pxqLpxq, so that σpxq
aPpBAq
ypxq “ Spxq . We may rewrite:
aPpBAq` px
Let us define σpxq “
bś
Kf pxq “ ´2ypxqf pxq `
where:
˛
Qf pxq “ ´
A
Qf pxq
,
Lpxq
ı
dξ ” Lpξq pV t0u q1 pξq ´ Lpxq pV t0u q1 pxq
` P t´1u px; ξq f pξq.
2iπ
x´ξ
26
(5.16)
(5.17)
p1q
For any f P H2 pAq, Qf is analytic in CzA, with singularities only where pV t0u q1 pξq has singularities,
p1q
in particular it is holomorphic in the neighborhood of A. It is clear that Im K Ď H1 pAq. Let ϕ P Im K,
p1q
and f P H2 pAq such that ϕ “ Kf . We can write:
˛
dξ
dξ σpξq f pξq
σpxq f pxq “ Res
σpξq f pξq “ ψpxq ´
,
(5.18)
ξÑx ξ ´ x
ξ´x
A 2iπ
where:
ψpxq “ ´ Res
ξÑ8
dξ
σpξq f pξq.
ξ´x
(5.19)
Since f pxq P Op1{x2 q, ψpxq is a polynomial in x of degree at most g ´ 1. We then pursue the
computation:
˛
Qf pξq ¯
dξ
1
σpξq ´
´ ϕpξq `
σpxqf pxq “ ψpxq ´
2iπ ξ ´ x 2ypξq
Lpξq
˛A
”
ı
dξ
1
1
“ ψpxq `
Lpξq ϕpξq ` pQf qpξq
2iπ ξ ´ x 2Spξq
˛A
1
Lpξq
dξ
ϕpξq,
(5.20)
“ ψpxq `
2iπ
ξ
´
x
2Spξq
A
p1q
using the fact that S has no zeroes on A. Let us denote G : Im K Ñ H2 pAq the linear operator
defined by:
˛
dξ
1
Lpξq
1
rGϕspxq “
ϕpξq.
(5.21)
σpxq A 2iπ ξ ´ x 2Spξq
One also obtains:
f pxq “
ψpxq
` pG ˝ Kqrf spxq.
σpxq
(5.22)
p and its inverse
The extended operator K
5.2.2
It was first observed in [Ake96] that ψpxqdx
σpxq defines a holomorphic 1-form on the Riemann surface
ś
2
Σ : σ “ aPpBAq px ´ αpaqq. The space H 1 pΣq of holomorphic 1-forms on Σ has dimension g if all
αpaq are pairwise distinct (which is the case by offcriticality) and the number of cuts is pg ` 1q and.
So, if g ě 1, K is not invertible. But we can define an extended operator:
´
Since
xj´1 dx
σpxq
¯
0ďjďg´1
p : Hp1q pAq
K
2
ÝÑ
Im K ˆ Cr
f
ÞÝÑ
pKf, LA rf dxsq.
(5.23)
are independent, they form a basis of H 1 pΣq. On the other hand, the family
´˛
of linear forms:
˛
LA “
¯
,...,
A1
(5.24)
Ag
are independent, hence they determine a unique basis $h pxq “
ψh pxqdx
σpxq
P H 1 pΣq so that:
˛
@h, h1 P v1, gw,
$h1 pxq “ δh,h1 .
(5.25)
Ah
LA thus induces a linear isomorphism of the space of H 1 pΣq. Its inverse can be written:
L´1
A rws “
g
ÿ
h“1
27
wh $h pxq
(5.26)
p is an isomorphism, its inverse being given by:
We deduce that K
“
‰
´1
L
w
´
L
rpGϕqdxs
pxq
A
A
´1
p rϕ, wspxq “
K
` Gϕpxq,
dx
(5.27)
p ´1 ϕ “ K
p ´1 rϕ, ws. The continuity of this
where G is defined in (5.21). We will use the notation K
w
inverse operator is the key ingredient of our method:
p1q
Lemma 5.1 Im K is closed in H2 pAq, and there exists s constant kΓ pAq, kΓ1 pAq ą 0 such that:
@pϕ, wq P Im K ˆ Cg ,
p ´1 ϕk ď kΓ pAq kϕk ` k 1 pAq|w|
kK
w
Γ
Γ
Γ
(5.28)
l
Remark 1. If one is interested in controlling the large N expansion of the correlators explicitly in
p ´1 . For this
terms of the distance of xi ’s to A, it is useful to give an explicit bound on the norm of K
w
purpose, let δ0 ą 0 be small enough but fixed once for all, and we move the contour in (5.21) to a
contour close staying at distance larger than δ0 from A. If we choose now a point x so that dpx, Aq ă η,
we can write:
˛
˛
dξ Lpξq
dξ Lpξq ϕpξq
ϕpxq
1
1
ϕpxq
´
`
(5.29)
Gϕpxq “
2Spxqσpxq σpxq dpξ,Aq“δ0 2iπ 2Spξq x ´ ξ
σpxq dpξ,Aq“δ0 2iπ 2Spξq x ´ ξ
Hence, there exist constants C, C 1 ą 0 depending only on the position of the pairwise disjoint segments
Ah such that, for any δ ą 0 smaller than δ0 {2:
kGϕkδ ď pCDc pδq ` C 1 q δ ´1{2 kϕkδ ` δ ´1{2 kϕkδ0
We set:
Dc pδq “
ˇ Lpξq ˇ
ˇ
ˇ
sup ˇ
ˇ
dpξ,Aq“δ Spξq
(5.30)
(5.31)
For δ small enough but fixed, Dc pδq blows up when the parameters of the model are tuned to achieve
a critical point, i.e. it measures the distance to criticality. Besides, we have for the operator L´1
A
written in (5.26):
U
max1ďhďg kψh k8
kL´1
rwsk
ď
|w|,
(5.32)
A
δ
inf dpξ,Aq“δ |σpxq|
and the denominator behaves like δ ´1{2 when δ Ñ 0. We then deduce from (5.27) the existence of a
constant C 2 ą 0 so that:
p ´1 ϕk ď pCDc pδq ` C 1 qδ ´κ kϕk ` δ ´κ |w|,
kK
w
δ
δ
(5.33)
with exponent κ “ 1{2.
Remark 2. From the expression (5.27) for the inverse, we observe that, if ϕ is holomorphic in CzS,
p ´1 ϕ for any w P Cg , in other words K
p ´1 pIm K X Hp1q pSqq Ď Hp1q pSq.
so is K
w
w
1
2
5.2.3
Other linear operators
Some other linear operators appear naturally in the Schwinger-Dyson equation. We collect them
below. Let us first define:
∆´1 W1 pxq
∆´1 P px; ξq
∆0 V pxq
t´1u
“ N ´1 W1 pxq ´ W1
pxq,
˛
dη
“
2L2 px; ξ, ηq∆´1 W1 pηq,
2iπ
A
“ V pxq ´ V t0u pxq.
28
(5.34)
(5.35)
(5.36)
Let also h1 , h2 be two holomorphic functions in U. We define:
˛
dξ L2 px; ξ, ξq
p1q
p1q
L1 : H1 pAq Ñ H2 pAq
f pξq ,
L1 f pxq “
2iπ
Lpxq
˛A
dξ1 dξ2 L2 px; ξ1 , ξ2 q
p2q
p1q
L2 : H1 pAq Ñ H1 pAq
f pξ1 , ξ2 q ,
L2 f pxq “
2
Lpxq
A p2iπq
˛
f pξq
dξ Lpξq
p1q
p2q
,
Mx1 : H1 pAq Ñ H1 pAq
Mx1 f pxq “
1 ´ ξq2
2iπ
Lpxq
px
´
ξqpx
A
˛
¯
dξ ´ Lpξqh1 pξq
1
p1q
p1q
Nh1 ,h2 : H1 pAq Ñ H1 pAq
` h2 pξq f pξq ,
Nh1 ,h2 f pxq “
Lpxq A 2iπ
x´ξ
p1q
p1q
∆K : H1 pAq Ñ H1 pAq
p∆Kqf pxq “ ´Np∆0 V q1 ,∆´1 P px;‚q rf spxq ` 2∆´1 W1 pxq f pxq
2¯
1´
1´
pBx ` L1 qf pxq.
(5.37)
`
N
β
All those operators are continuous for appropriate norms, since we have the bounds, for δ0 small
enough but fixed, and δ small enough:
8
kL1 f pxqkδ
ď
C kL2 kU
kf kδ0 ,
DL pδq
kL2 f pxqkδ
ď
C 2 kL2 kU
kf kδ0 ,
DL pδq
kMx1 f kδ
ď
CkLkU
kf kδ{2 ,
DL pδq δ 3
kNh1 ,h2 kδ
ď
kh1 kU kf kδ ` C
kp∆Kqf kδ
ď
8
8
8
8
kLh1 kU ` kh2 kU
kf kδ0 ,
δ0 DL pδq
ˇ
`
˘
2 ˇˇ 2C
ˇ
8
kf kδ{2
kp∆0 V q1 kU ` 2 k∆´1 W1 kδ kf kδ ` ˇ1 ´ ˇ
β N δ2
8
8
kL p∆0 V q1 kU ` k∆´1 P kU2
`C
kf kδ0
DL pδq δ0
8
(5.38)
for any f in the domain of definition of the corresponding operator, and:
C “ `pAq{π ` pg ` 1q,
DL pδq “
inf
dpx,Aqěδ
|Lpxq|.
(5.39)
If all edges are soft, DL pδq ” 1, whereas if there exist at least one hard edge, DL pδq scales like δ when
δ Ñ 0.
5.3
5.3.1
Recursive expansion of the correlators
Rewriting Schwinger-Dyson equations
For n ě 2 and m P v0, n ´ 1w, we can organize the Schwinger-Dyson equation of Corollary 4.2 as
follows:
”
1´
2¯ ı
K ` ∆K `
1´
Bx Wn|m px, xI q “ An`1|m ` Bn|m ` Cn´1|m ` Dn´1|m ,
(5.40)
N
β
29
where:
An`1|m px; xI q
Bn|m px; xI q
“ N ´1 pL2 ´ idqWn`1|m px, x, xI q,
ˆ ˙
!
)
ÿ
m
´1
1 |m1 px, xJ q b Wn´|J|´m1 |m´m1 px, xIzJ q ,
“ N pL2 ´ idq
W
|J|`1`m
m1
1
JĎI, 0ďm ďm
pJ,m1 q‰pH,0q,pI,mq
Cn´1|m px; xI q
“
Dn´1|m px; xI q
“
2 ÿ
Mxi Wn´1|m px, xIztiu q,
βN iPI
ÿ Lpaq
2
Ba Wn´1|m pxI q.
βN
x´a
´
(5.41)
aPpBAq`
p ´1 ϕ “ K
p ´1 rϕ, ws and by definition (4.5) of Wn|m ,
This equation can be rewritten, with the notation K
w
”
p ´1
Wn|m px, xI q “ K
Wn|m`1 pxI q An`1|m px, xI q ` Bn|m px, xI q ` Cn´1|m px, xI q ` Dn´1|m px, xI q
ı
2¯
1´
1´
pBx ` L1 qWn|m px, xI q , (5.42)
´ p∆KqrWn|m px, xI qs ´
N
β
where it is understood that the operators all act on the first variable (or the two first variables for
L2 ).
For n “ 1 and m “ 0, we have almost the same equation; with the notation of (5.34), and in view
of (4.2),
” A pxq ` D
1´
2¯
2
0
p ´1
∆´1 W1 pxq “ K
´
1
´
pBx ` L1 qW1 pxq
∆´1 W1|1
N
N
β
`
˘2 ı
´ Np∆0 V q1 ,0 W1 pxq ´ ∆´1 W1 pxq ,
(5.43)
where:
t´1u
∆´1 W1|1 “ LA r∆´1 W1 s “ N ´1 LA rW1 s ´ LA rW1
s
(5.44)
is the first correction to the expected filling fractions.
We would like ∆K to be negligible compared to K in (5.40), that is
p w p∆Kf qk ! kf k
kK
δ
δ
(5.45)
This can be controlled thanks to (5.33) and (5.38). From our assumptions, we know that:
8
8
k∆0 V kU P Op1{N q.
k∆´1 P kU2 P op1q,
(5.46)
Taking into account those estimates in (5.38), we observe that it will be possible independently of the
´1{2
nature of the edges provided δ is restricted to be larger than δN such that limN Ñ8 δN k∆´1 W1 kδN “
0, and limN Ñ8 1´2{β
5{2 “ 0. Given the a priori bound in Corollary 3.7 on ∆´1 W1 , this can be realized
N δN
independently of β if:
?
ln N ln δN
´κ
lim
Dc pδN q δN
“0
(5.47)
N Ñ8
N
δN
Since we consider here a fixed, off-critical potential, Dc pδq remains bounded, and since κ “ 1{2, this
condition is equivalent to:
´ ln N ¯1{3 ln1{3 δ
N
lim
“0
(5.48)
N Ñ8
N
δN
It justifies the introduction of the sequence δN in Definition 5.4. Then, by similar arguments, kDn|m kδ
will always be exponentially small when N is large provided δ ě δN , and thus negligible in front of
the other terms.
c
30
5.3.2
Initialization and order of magnitude of Wn
We remind θ “ 1 and κ “ 1{2 here. The goal of this section is to prove the following bounds.
t0u
Proposition 5.2 There exists a function W1
t´1u
W1 “ N W 1
t0u
` W1
` ∆ 0 W1 ,
p1q
P H2 pSq independent of N so that:
d
¯
´ ln3 N ´ D pδq ¯2
c
δ ´p2θ`κq
∆ 0 W1 P O
N
DL pδq
(5.49)
It is given by:
t0u
W1 pxq
p ´1t0u
“K
W1|1
"”
*
´
ı
2¯
t´1u
´ 1´
pBx ` L1 q ´ NpV t1u q1 ,0 W1
pxq
β
tn´2u
Proposition 5.3 For any n ě 2, there exists a function Wn
tn´2u
Wn|m “ N 2´n pWn|m
where:
(5.50)
pnq
P H2 pSq so that:
` ∆n´2 Wn|m q
¯
´
´ D pδq ¯3n´3
c
δ ´θpn´mq´pκ`θqp3n´3q
∆n´2 Wn P O N ´1{2 pln N q2n´3{2
DL pδq
(5.51)
(5.52)
Prior to those results, we are going to prove the following bound:
Lemma 5.4 Denote rn˚ “ 3n ´ 4. For any integers n ě 2 and r ě 0 such that r ď rn˚ , we have:
´ n´r
¯
´ D pδq ¯r
n`r
c
Wn|m P O N 2 pln N q 2
δ ´θpn´mq´pκ`θqr
DL pδq
(5.53)
Proof. The a priori control of correlators (3.28) provides the result for r “ 0. Let s be an integer,
and assume the result is true for any r P v0, sw. Let n be such that s ` 1 ď rn˚ “ 3n ´ 4. We consider
(5.42) which gives Wn|m in terms of Wn`1|m and Wn1 |m1 for n1 ă n or n1 “ n but m1 ą m, and we
p We obtain the following bound on the A-term:
exploit the control (5.33) on the inverse of K.
´ n´ps`1q
´ D pδq ¯s`1
¯
n`s`1
c
´pn´mqθ´ps`1qpκ`θq
´1{2
p ´1
2
2
K
pA
q
P
O
N
pln
N
q
δ
`
kW
k
δ
,
n`1|m
n|m`1 δ
Wn|m`1
DL pδq
(5.54)
and we now argue that it will be the worse estimate among all other terms, given that δ ě δN . The
B-term involves linear combinations of Wj`1|m1 b Wn´j|m´m1 . Notice that:
˚
˚
˚
s ´ rj`1
ď rn˚ ´ rj`1
“ rn´j
(5.55)
˚
1
Therefore, we can use the recursion hypothesis with r “ rj`m
1 `1 “ 3pj `m q´1 to bound Wj`m1 `1|m1 ,
˚
and with r “ s ´ rj`m
1 `1 to bound Wn´j´m1 |m´m1 , and we find:
´ D pδq ¯s`1
¯
´ n´ps`1q
n`ps`1q
c
´pn´mqθ´ps`1qpκ`θq
´κ
p ´1
2
2
pln
N
q
δ
`
kW
k
δ
K
pB
q
P
O
N
n|m`1
n|m
Wn|m`1
δ
DL pδq
(5.56)
˚
which is of the same order as (5.54). The C-term involves Wn´1|m . If s ď rn´1 , we can use the
recursion hypothesis at r “ s to bound it as
p ´1
K
Wn|m`1 pCn´1|m q P O
´ δ 2θ´3
´ D pδq ¯s`1
¯
n´ps`1q
n`1`s
c
´pn´mqθ´ps`1qpκ`θq
2
2
N
pln
N
q
δ
N 2 ln N
DL pδq
31
(5.57)
˚
The prefactor makes this term negligible comparing to (5.54). If s ą rn´1
, we can only use the
˚
recursion hypothesis for r “ rn´1
, and find the bound:
p ´1
K
Wn|m`1 pCn´1|m q
´
´ D pδq ¯3n´6
c
P O N 2´n pln N q2n´4
δ ´pn´mqθ´p3n´4qpκ`θq`2pθ´κq
DL pδq
¯
`kWn|m`1 kδ δ ´κ ,
(5.58)
and thus still negligible compared to the A-term when N Ñ 8 and δ Ñ 0. We can conclude by
recursion from m “ n to m “ 0, taking into account the assumed expansion (5.3) for Wn|n .
l
Proof of Proposition 5.2. Lemma 5.4 for r “ 1 gives the bound:
´a
W2 P O
Then, in (5.43), we find that A2 P O
`
N ln3 N
W2
N DL pδq
Dc pδq ´p3θ`κq ¯
δ
DL pδq
(5.59)
˘
, and we already know that |∆´1 W1 | goes to zero
2
p ´1
for uniform convergence in any compact subset of CzA, so that }K
Wn|m`1 p∆´1 W1 q lδ is neglectable
p ´1 , we deduce
with respect to ∆´1 W1 at list for δ not going to zero with N . By continuity of K
t0u
that N ∆´1 W1 pxq has a limit W1 for convergence in any compact subset of CzA, given by (5.50).
p1q
t´1u
Reminding Remark 2 page 28, this limit belongs to H2 pSq, and given the behavior of W1
at the
edges, we have a naive bound:
´ δ ´3{2 ¯
t0u
,
(5.60)
pBx ` L1 qW1 P O
DL pδq
and we have already argued at the end of § 5.3.1 that kN ´1 pBx ` L1 qp∆´1 W1 qkδ was negligible
compared to k∆´1 W1 kδ{2 provided δ ě δN . Moreover, Therefore, the worse estimate on the error
p ´1 given in (5.33), we find:
∆0 W1 is given by the term A2 . Taking into account the effect of K
∆0 W 1 P O
´ pln N q3{2 ´ D pδq ¯2
¯
c
?
δ ´p3θ`2κq
DL pδq
N
which is the desired result.
(5.61)
l
Proof of Proposition 5.3 We already know the result for n “ 1. Let n ě 2, m P v0, n ´ 1w, and
assume the result holds for all n1 P v1, n ´ 1w and m1 P vm, nw. We want to use (5.40) once more to
compute Wn|m . Applying Lemma 5.4 to r “ 3n ´ 4 for n ě 2, we find:
´
˘´p3n´4q ´pn`1´mqθ´p3n´4qpθ`κq ¯
pln N q2n´3{2 `
?
DL pδq
δ
,
An`1|m P O N 2´n
N
(5.62)
whereas the recursion hypothesis implies that Bn|m and Cn´1|m are of order OpN 2´n q. Hence,
tn´2u
Wn|m “ N 2´n pWn|m ` ∆n´2 Wn|m q with:
tn´2u
Wn|m px, xI q
“
p ´1tn´2u
K
”
Wn|m`1 pxI q
!
`pL2 ´ idq
2ÿ
tn´3u
Mxi Wn´1|m px, xI q
(5.63)
β iPI
)ı
ÿ ˆ m ˙ t|J|´1`m1 u
tn´|J|´2´m1 u
W
px,
x
q
b
W
px,
x
q
1
1
1
1
J
IzJ
|J|`1`m |m
n´|J|´m |m´m
m1
1
´
0ďm ďm
JĎI
The error term ∆n´2 Wn|m receives contribution either from errors ∆n1 ´2 Wn1 |m1 appearing in Bn|m
2
and Cn´1|m , which we already know how to bound. And the restriction that limN Ñ8 N δN
“ `8
32
guarantees that they are negligible in front of An`1|m computed in (5.62). Hence, we deduce by
tn´2u
substracting Wn|m and inverting K that:
∆n´2 Wn|m
c
´ pln N q4n´3 ´ D pδq ¯3n´3
¯
c
PO
δ ´pn´mqθ´p3n´3qpθ`κq
N
DL pδq
(5.64)
which is the desired result at step pn, mq. We conclude by recursion.
5.4
l
Recursive expansion of the correlators
Proposition 5.5 For any k0 ě 0, we have for any n ě 1:
Wn|m px1 , . . . , xn q “
k0
ÿ
tku
N ´k Wn|m px1 , . . . , xn q ` N ´k0 p∆k0 Wn|m qpx1 , . . . , xn q,
(5.65)
k“n´2
where:
tku
pn´mq
piq for any n ě 1 and any k P v0, . . . , k0 w, Wn|m has a limit when N Ñ 8 in H2
pointwise convergence in any compact of pCzAqn´m , and:
´ D pδq ¯n`2k
¯
´
n`k
c
tku
Wn|m P O pln N q 2
δ ´pn´mqθ´pn`2kqpκ`θq .
DL pδq
pn´mq
piiq for any n ě 1, ∆k0 Wn|m P H2
∆k0 Wn|m P O
pSq for
(5.66)
pAq and:
´ pln N qn`k0 `1{2 ´ D pδq ¯n`2k0 `1
¯
c
?
δ ´pn´mqθ´pn`2k0 `1qpκ`θq .
DL pδq
N
(5.67)
Proof. The case k0 “ 0 follows from § 5.3.2, and we prove the general case by recursion on k0 , which
can be seen as the continuation of the proof of Proposition 5.3. Assume the result holds for some
k0 ě 0. Let us decompose:
V “
kÿ
0 `2
N ´k V tku ` N ´pk0 `2q ∆k0 `2 V.
(5.68)
k“0
We already know that the loop equations are satisfied up to order N 1´k0 . We can decompose the
remainder as:
!
˘
` tk0 u
1´
2¯ )
tk0 u
N ´pk0 ´1q K ` ∆K `
1´
Bx ∆k0 Wn|m px, xI q “ N ´k0 En|m
px; xI q ` Rn|m
px; xI q . (5.69)
N
β
It is understood that all linear operators appearing here (and defined in § 5.2) act on the variable(s)
x. We have set:
tk u
0
En|m
px; xI q
“
“ tk0 ´1u
‰
pL2 ´ idq Wn`1|m
px, x, xI q
ÿ ÿ ˆm˙
“ tku
‰
tk0 ´ku
`
pL2 ´ idq W|J|`1`m1 |m1 px, xJ q b Wn´|J|´m
1 |m´m1 px, xIzJ q
1
m
0ďkďk JĎI
0
0ďm1 ďm
k0
´
ÿ
“ tk0 u
‰
“ tku
‰
2¯
´ 1´
pBx ` L1 q Wn|m px, xI q `
NpV tk0 `1´ku q1 ,0 Wn|m px, xI q
β
k“n0 ´2
‰
“ tk0 u
2ÿ
´
Mxi Wn´1|m px, xIztiu q ,
(5.70)
β iPI
33
and:
tk u
0
Rn|m
px; xI q
“
‰
pL2 ´ idq ∆k0 Wn`1|m px, x, xI q
ÿ
“`
˘
‰
pL2 ´ idq ∆k0 W|J|`1`m1 |m1 px, xJ q b Wn´|J|´m1 |m´m1 px, xIzJ q
`
“
0ďm1 ďm
JĎI
ÿ
ÿ
`
“ tk0 ´ku
`
˘‰
pL2 ´ idq W|J|`1`m
1 |m1 px, xJ q b ∆k Wn´|J|´m1 |m´m1 px, xIzJ q
0ďkďk0 JĎI
0ďm1 ďm
k
ÿ
`
“
‰
“ tk ´lu
‰ 2ÿ
Mxi ∆k0 Wn´1|m px, xIztiu q
Np∆l`1 V q1 ,0 Wn|m0 px, xI q ´
β
iPI
l“´1
´
2
β
ÿ
aPpBAq`
Lpaq
Ba Wn´1|m pxI q.
x´a
Let us denote:
tku
wn|m pN, δq “
pln N qn`k`1{2 ´ Dc pδq ¯n`2k`1 ´pn´mqθ´pn`2k`1qpκ`θq
?
δ
.
DL pδq
N
(5.71)
p from (5.33), we
Thanks to the recursion hypothesis and the bound on the norm of the inverse of K
find:
`
˘
p ´1 Rn|m P O wtk`1u pN, δq .
K
(5.72)
n
0
This bound arise from the three first lines. Indeed, the two last terms are negligible compared to
tk`1u
as noticed in the proof of Lemma 5.4, and the term involving ∆l`1 V is of order pln N qp {N for
wn
tk`1u
. Therefore, we
some p, with a dependence in δ which is less divergent than that appearing of wn
deduce by recursion on m from m “ n to m “ 0 that N ∆k0 Wn|m converges to:
“ tk u
‰
tk `1u
p ´1tk `1u
Wn|m0 px, xI q “ K
En 0 px, xI q ,
(5.73)
0
Wn|m`1 pxI q
pointwise and uniformly on any compact of pCzAqn´m . Besides, the estimate (5.72) yields the bound
(5.67) for the error, while the recursion hypothesis combined with (5.73) leads to (5.66).
l
This proves the first part of Theorem 1.3 for real-analytic potentials (i.e. the stronger Hypothesis 1.3 instead of 1.4). For given n and k, the bound on the error ∆k Wn depend only on a finite
1
tk1 u
number of constants v tk u , wn1 appearing in Hypotheses 5.1.
5.5
Central limit theorem
With Proposition 5.2 at our disposal, we can already establish a central limit theorem for linear
statistics of analytic functions in the fixed filling fraction model. It will be refined for non-analytic
but smooth enough functions in § 6.1.
Proposition 5.6 Assume the result of Proposition 5.2. Let ϕ : A Ñ R extending to a holomorphic
function in a neighborhood of S. Then:
N
”
´ÿ
¯ı
´
¯
1
;A
µVN,‚,β
exp
ϕpλi q “ exp N Lrϕs ` M rϕs ` Qrϕ, ϕs ` op1q ,
2
i“1
˛
where:
Lrϕs “
A
t0u
W1
˛
dξ
t´1u
ϕpξq W1
pξq,
2iπ
M rhs “
A
dξ
t0u
ϕpξq W1 pξq
2iπ
has been introduced in (5.49), and Q is a quadratic form given in (5.78) or (5.79) below.
34
(5.74)
(5.75)
2t
Proof. Let us define Vt “ V ´ βN
ϕ. Since the equilibrium measure is the same for Vt and V , we
still have the result of Proposition 5.2 for the model with potential Vt for any t P r0, 1s, with uniform
errors. We can thus write:
ˆ 1 ˛
N
”
´ÿ
¯ı
dξ
V ;A
ln µN,‚,β exp
t ϕpλi q
“
dt
W1Vt pξq ϕpξq
2iπ
0
A
i“1
ˆ 1 ˛
“
‰
dξ
V ;t´1u
V ;t0u
dt
“
ϕpξq N W1 t
` W1 t pξq ` op1q (5.76)
0
A 2iπ
V ;t0u
As already pointed out, W1 t
V ;t0u
W1 t
V ;t0u
“ W1
, and from (5.49):
V ;t0u
“ W1
´
˘ V ;t´1u
2t ` p ´1
s
K t0u ˝ Nϕ1 ,0 rW1
W1|1
β
(5.77)
Hence (5.74), with:
2
Qrϕ, ϕs “ ´
β
˛
A
` ´1
˘
dξ
p t0u ˝ Nϕ1 ,0 rW V ;t´1u spξq
ϕpξq K
1
W1|1
2iπ
If we restrict to the model with fixed filling fraction, it can be simplified to:
‹
dξ1 dξ2
V ;t0u
Qrϕ, ϕs “
ϕpξ1 qϕpξ2 q W2
pξ1 , ξ2 q
2
p2iπq
A
(5.78)
(5.79)
V ;t0u
V
where W2
has been introduced in (5.51) and we recall W2|1
“ 0 for the model with fixed filling
fractions. From the proof of Proposition 5.2, we observe that the op1q in (5.74) is uniform in h such
l
that supξPΓE |ϕpξq| is bounded by a fixed constant.
řN
In other words, the random variable Φ “ i“1 ϕpλi q ´ Lrϕs converges almost surely to a Gaussian
variable with mean M rϕs and variance Qrϕ, ϕs. This is a generalization of the central limit theorem
already known in the one-cut regime [Joh98, BG11]. A similar result was recently obtained in [Shc12].
In the next Section, we are going to extend it to holomorphic h which could be complex-valued on A
(Proposition 6.3). In general, to establish the central limit theorem, one could be tempted to use the
definition of the correlators:
˛ ź
n
N
¯ıˇ
”
´ÿ
dξi
ˇ
V ;A
n
t ϕpλi q ˇ
“
ϕpξi q WnV pξ1 , . . . , ξn q,
(5.80)
Bt ln µN,‚,β exp
2iπ
t“0
n
A i“1
i“1
;A
then represent GN ptq “ ln µVN,‚,β
retΦ s by its Taylor expansion up to t “ 1, and use the result of
V
2´n
Proposition 5.2 that Wn P OpN
q to conclude. However, for any fixed N , GN ptq is analytic in the
;A
domain of the complex plane where µVN,‚,β
retΦ s does not vanish, and it is not obvious that for N large
enough (although it will turn out to be true) that this does not happen for some t0 P C with |t0 | ă 1,
i.e. that the Taylor series converges in the appropriate domain.
6
Fixed filling fraction: refined results
In this section, we show how the asymptotic expansion of multilinear statistics for non-analytic test
functions can be deduced from our results, thanks to their explicit dependence on the distance of the
variables x (appearing in the correlators) to A. We also show how to extend our results to the case of
harmonic potentials, and potentials containing a complex-valued term of order Op1{N q. The latter is
performed by using fine properties of analytic functions (the two-constants theorem) as was recently
proposed in [Shc12].
35
6.1
Multilinear statistics for non-analytic test functions
Our methods establish a control on n-point correlators Wn px1 , . . . , xn q, depending on how x1 , . . . , xn
approach the range of integration A. We now argue that it gives a control n-linear statistics for
test functions with regularity lower than analytic. If s is a finite-dimensional vector, we denote
ř
|s|1 “ i |si |.
Lemma 6.1 Let fn be a holomorphic function defined in a neighborhood of An in pCzAqn . Assume
there exists C, r, and η P p0, 1q small enough, such that
@δ ě η,
sup
|fn pξ1 , . . . , ξn q| ď
dpξi ,Aqěδ
C
δr
(6.1)
Then, there exists a constant C 1 so that, for any s satisfying |s|1 P rr, r{ηs, we have:
ˇ˛
ˇ
ˇ
n
ˇ
ź
dξj isj ξj
ˇ
e
fn pξ1 , . . . , ξn qˇ ď C 1 |s|r1
An j“1 2iπ
(6.2)
Proof. For δ small enough but larger than η, let Cpδq be the contour surrounding A such that
dpξ, Aq “ δ for any ξ P Cpδq. When A has pg`1q connected components, its length is 2p`pAq`pg`1qπδq.
For any s P R, we find:
ˇ˛
ˇ
ˇ
n
ˇ
ź
dξj ´isj ξj
ˇ
e
fn pξ1 , . . . , ξn qˇ “
2iπ
n
A j“1
ď
ˇ˛
ˇ
ˇ
ˇ
dξj ´isj ξj
ˇ
e
fn pξ1 , . . . , ξn qˇ
2iπ
n
C pδq
´ `pAq
¯n
C
` pg ` 1qδ
e|s|1 δ´r ln δ
π
(6.3)
We now optimize this inequality keeping |s|1 large in mind, by choosing δ “ r{|s|1 , which leads to the
desired result.
l
Corollary 6.2 Let ϕ : Rn Ñ C be a continuous function with compact support, so that its Fourier
transform satisfies:
p P op|s|´r q,
ϕpsq
|s| Ñ 8
(6.4)
Then, for any integer k0 such that r ě 1 ` κ ` 2θ `
;A
µVN,β
n ´ ÿ
N
”ź
j“1
k0
ÿ
¯ı
ϕpλij q
ij “1
c
“
2k0
n pκ
` θq, we have an expansion of the form:
` ´pk `1{2q
˘
0
N ´k Mtku
pln N qn`k0 `1{2
n rϕs ` o N
(6.5)
k“n´2
pη psq “ e´η|s| ϕpsq.
p
It is analytic
Proof. Let η ą 0, and define a function ϕη by its Fourier transform ϕ
in the strip tξ P C, |Im ξ| ă ηu, and we may write:
;A
µVN,β
n ´ ÿ
N
”ź
j“1
ij “1
¯ı
ϕη pλij q
c
˛
“
“
n
¯
´ź
dξj
ϕη pξj q Wn pξ1 , . . . , ξn q
(6.6)
An j“1 2iπ
ˆ ´ź
n
n
¯˛ ´ ź
dξj ´isj ξj ¯
p jq
Wn pξ1 , . . . , ξn q
dsj e´η|sj | ϕps
e
Rn j“1
An j“1 2iπ
We may insert the large N expansion of the correlators established in Proposition 5.5:
Wn pξ1 , . . . , ξn q “
k0
ÿ
N ´k Wntku pξ1 , . . . , ξn q ` N ´k0 ∆k0 Wn pξ1 , . . . , ξn q
k“n´2
36
(6.7)
where:
k∆k0 Wn kδ P O
´ pln N qn`k0 `1{2 ´ D pδq ¯n`2k0 `1
¯¯
c
?
δ ´nθ´pn`2k0 qpκ`θq
DL pδq
N
(6.8)
We may pass to the limit η Ñ 0 in (6.6) when the integrand in the right-hand side is integrable near
|s|1 “ 8. It constrains the allowed behavior for ϕpsq
p
at |s| Ñ 8. The worse behavior at |s|1 “ 8
comes from the error term ∆k0 Wn . Lemma 6.1 implies that, for any ε ą 0, there exists a constant
Cε ą 0 such that:
ˇ˛
ˇ
n
ˇ
ˇ
´ź
¯n`2k0 `1
dξj ´isj ξj ¯
pln N qn`k0 `1{2 ´ Dc p|s|´1
ˇ
ˇ
nθ`pn`2k0 qpκ`θq`ε
1 q
?
Wn pξ1 , . . . , ξn qˇ ď Cε
e
|s|1
ˇ
ˇ An j“1 2iπ
ˇ
DL p|s|´1
q
N
1
(6.9)
p P op|s|´r q. Then, integrability at |s|1 in (6.6) requires:
Assume now that ϕpsq
nθ ` pn ` 2k0 qpκ ` θq ´ npr ` εq ă ´n
(6.10)
In other words, performing an expansion up to opN ´k0 q if the regularity exponent r satisfies:
r ě 1 ` κ ` 2θ `
2k0
pκ ` θq.
n
(6.11)
l
6.2
Extension to harmonic potentials
The main use of the assumption that V is analytic came from the representation (1.6) of n-linear
statistics described by a holomorphic function, in terms of contour integrals of the n-point correlator.
If ϕ is holomorphic in a neighborhood of A, its complex conjugate ϕ is antiholomorphic, and we can
also represent:
N
”ÿ
ı ˛ dx
;A
µVN,β
ϕpxq W1 pxq
(6.12)
ϕpλi q “
A 2iπ
i“1
In this paragraph, we explain how to use a weaker set of assumptions than Hypothesis 1.3 , where
”analyticity” and ”1{N expansion of the potential” are weakened as follows.
Hypothesis 6.1
‚ (Harmonicity) V : A Ñ R can be decomposed V “ V1 ` V2 , where V1 , V2
extends to holomorphic functions in a neighborhood U of A.
‚ (1{N expansion of the potential) For j “ 1, 2, there exists a sequence of holomorphic functions
tku
tku
pVj qkě0 and constants pvj qk so that, for any K ě 0:
K
ˇ
ˇ
ÿ
ˇ
ˇ
tku
tKu
sup ˇVj pξq ´
N ´k Vj pξqˇ ď vj N ´pK`1q
ξPU
(6.13)
k“0
In other words, we only assume V to be harmonic. ”Analyticity” corresponds to the special case
V2 ” 0. The main difference lies in the representation (6.12) of expectation values of antiholomorphic
statistics, which come into play at various stages, but do not affect the reasoning. Below chronologically Section 5, we enumerate below the small changes to take into account.
In § 4, in the Schwinger-Dyson equations (Theorem 4.2 and 4.2), we encounter a term:
;A
µVN,β
N
n
N
”ÿ
¯ı
1
Lpλi q V 1 pλi q ź ´ ÿ
.
Lpxq x ´ λi j“2 i “1 xj ´ λij c
i“1
j
37
(6.14)
It is now equal to:
1
Lpxq
˛
A
dξ
V 1 pξq
1
Lpξq 1
Wn pξ, xI q ´
2iπ
x´ξ
Lpxq
˛
A
dξ
V 1 pξq
Lpξq 2
Wn pξ, xI q.
2iπ
x´ξ
(6.15)
Remark that (6.14) or (6.15) still defines a holomorphic function of x in CzA. The second line in
Corollary 4.3 has to be modified similarly. In § 5.2, we can define the operator K by (5.16) with Qpxq
now given by:
˛
dξ t´1u
Qf pxq “ ´
P
pξqpx; ξq f pξq
2iπ
A
˛
t0u
t0u
dξ LpξqpV1 q1 pξq ´ LpxqpV1 q1 pxq
f pξq
`
ξ´x
A 2iπ
˛
t0u
t0u
dξ LpξqpV2 q1 pξq ´ LpxqpV2 q1 pxq
`
f pξq.
(6.16)
ξ´x
A 2iπ
It is still a holomorphic function of x in a neighborhood of A, thus it disappears in the computation
leading to formula 5.22 for the inverse of K, which still holds. In § 5.2.3, the expression (5.37) for the
operator ∆K used in (5.40) should be replaced by:
1´
2¯
p∆Kqf pxq “ 2∆´1 W1 pxq f pxq `
1´
L1 f pxq
N
β
´Np∆0 V1 q1 ,∆´1 P px;‚q rf spxq ´ Np∆0 V2 q1 ,0 rf spxq,
(6.17)
and the bound (5.38) still holds, where v0 is replaced by v1,0 ` v2,0 introduced in (6.13). In § 5.3.15.4, all occurences of NV 1 ,0 rf spxq should be replaced by NpV1 q1 ,0 rf spxq ` NpV2 q1 ,0 rf spxq (and similarly
for Np∆k V q1 ,0 or NpV tku q1 ,0 ). The key remark is that the terms where V2 appear involve complex
tku
conjugates of contour integrals of the type gpξq Wn pξ, xI q or gpξq ∆k Wn pξ, xI q where g is some
holomorphic function in a neighborhood of A. Their norm can be controlled in terms of the norms
tku
of Wn or ∆k Wn on contours Γ, as were the terms involving V1 , so the recursive control of errors in
the 1{N expansion of correlators for the fixed filling fraction model is still valid, leading to the first
part of Theorem 1.3, and to the central limit theorem (Proposition 5.6) for harmonic potentials in a
neighborhood of A, which are still real-valued on A.
6.3
Complex perturbations of the potential
Proposition 6.3 The central limit theorem (5.74) holds for ϕ : A Ñ C, which can be decomposed as
ϕ “ ϕ ` ϕ2 , where ϕ1 , ϕ2 are holomorphic functions in a neighborhood of A.
Proof. We present the proof for ϕ “ t f , where t P C and f : A Ñ R extends to a holomorphic
function in a neighborhood of A. Indeed, the case of f : A Ñ R which can be decomposed as
f “ f1 ` f2 with f1 , f2 extending to holomorphic functions in a neighborhood of A, can be treated
similarly with the modifications pointed out in § 6.2. Then, if ϕ : A Ñ C can be decomposed
I
as ϕ “ ϕ1 ` ϕ2 with ϕ1 , ϕ2 holomorphic, we may decompose further ϕj “ ϕR
j ` iϕj , then write
2
R
I
I
Ṽ “ V ´ βN
pϕR
1 ` ϕ2 q and f “ pϕ1 ´ ϕ2 q, and:
N
N
N
”
´ÿ
¯ı
”
´ÿ
¯ı
”
´ÿ
¯ı
;A
;A
Ṽ ;A
R
µVN,,β
exp
hpλi q “ µVN,,β
exp
pϕR
`
ϕ
q
µ
exp
if
pλ
q
.
i
1
2
N,,β
i“1
i“1
(6.18)
i“1
The first factor can be treated with the initial central limit theorem (Proposition 5.6), while an
equivalent of the second factor for large N will be deduced from the following proof applied to the
potential Ṽ .
38
This proof is inspired from that of [Shc12, Lemma 1]. From Theorem 1.3 applied to V up to
tku
op1q, we introduce Wn, for pn, kq “ p1, ´1q, p2, 0q, p1, 0q (see (5.51)-(5.49)). If t P R, the central limit
theorem (Proposition 5.6) applied to ϕ “ t f implies:
;A
µVN,‚,β
N
”´ ÿ
¯ı
t f pλi q “ GN ptqp1 ` RN ptqq,
i“1
¯
´
t2
GN ptq “ exp N t Lrf s ` t M rf s ` Qrf, f s , (6.19)
2
where suptPr´T0 ,T0 s |RN ptq| ď CpT0 q ηN and limN Ñ8 ηN “ 0. Let T0 ą 0, and introduce the function:
R̃N ptq “
1
RN ptq.
CpT0 qηN
(6.20)
For any fixed N , it is an entire function of t, and by construction
sup
|R̃N ptq| ď 1.
(6.21)
tPr´T0 ,T0 s
Besides, for any t P C, we have
N
N
ˇ
¯ıˇ
”
´ÿ
¯ı
”
´ÿ
ˇ
ˇ V ;A
;A
t f pλi q ˇ ď µVN,‚,β
exp
pRe tq f pλi q
ˇµN,‚;β exp
(6.22)
i“1
i“1
Therefore, we deduce that
sup |R̃N ptq|
ď
|t|ďT0
ď
ď
1
GN pRe tq
sup
CpT0 qηN |t|ďT0 |GN ptq|
´ pIm tq2
¯
1
sup exp
Qrf, f s
CpT0 qηN |t|ďT0
2
1
C 1 pT0 qηN
(6.23)
for some constant C 1 pT0 q. By the two-constants lemma [NN22], (6.21)-(6.23) imply
@T Ps0, T0 r,
sup |R̃N ptq| ď pC 1 pT0 qηN q´2φpT,T0 q{π ,
|t|ďT
´
φpT, T0 q “ arctan
2T {T0 ¯
. (6.24)
1 ´ pT {T0 q2
In particular, for any compact K of the complex plane, we can find an open disk of radius T0 which
contains K, and thus show (6.19) with RN ptq P op1q uniformly in K.
l
We observe from the proof that Proposition 6.3 cannot be easily extended to TN |t| P Op1q with
TN Ñ `8. Indeed, the ratio GN pTN pRe tqq{|GN pTN tq| in (6.23) will not be bounded when N Ñ 8,
hence applying the two-constants lemma as above does not show RN ptq Ñ 0.
Corollary 6.4 In the model with fixed filling fractions , assume the potential V0 satisfies Hypotheses 5.1. Then, if ϕ : A Ñ C can be decomposed as ϕ “ ϕ1 ` ϕ2 with ϕ1 , ϕ2 extending to holomorphic functions in a neighborhood of A, then the model with fixed filling fractions and potential
V “ V0 ` ϕ{N satisfies Hypotheses 5.1. Therefore, the result of Proposition 5.5 also holds: the
correlators have a 1{N expansion.
Proof. Hypothesis 5.1 contrains only the leading order of the potential, i.e. it holds for pV0 , q iff it
holds for pV “ V0 `h{N, q. Proposition 6.3 implies a fortiori the existence of constants C` , C´ , C ą 0
such that:
V ;A
C´ C N ď |ZN,,β
| ď C` C N
(6.25)
39
Using this inequality as an input, we can repeat the proof given in Section 3 to check to obtain Corollary 3.7 (i.e. the a priori control reminded in (5.9)-(5.10)) for the potential V . Then, in the recursive
analysis of the Schwinger-Dyson equation of Section 5 for the model with fixed filling fractions, the
fact that the potential is complex-valued does not matter, so we have proved the 1{N expansion of
the correlators.
l
This proves Theorem 1.3 in full generality.
6.4
1{N expansion of n-kernels
We can apply Corollary 6.4 to study potentials of the form:
Vc,x pξq “ V ´
2 ÿ
cj lnpxj ´ ξq
βN j
(6.26)
where xj P CzA, and thus derive the asymptotic expansion of the kernels in the complex plane, i.e.
řn
řN
Corollary 1.7 and 1.8. Indeed, let us introduce the random variable Hc pxq “ j“1 cj i“1 lnpxj ´ λi q.
“
‰
;A
We now know from Proposition 6.3 that ln µVN,,β
etHc pxq is an entire function. Therefore, its Taylor
series is convergent for any t P C, and we have:
´
¯
;A “ tHc pxq ‰
Kn,c pxq “ exp ln µVN,,β
e
n
n
¯
¯
´ÿ 1 ˛ ź
dξi ´ ÿ
cj lnpxj ´ ξi q Wr pξ1 , . . . , ξr q
(6.27)
“ exp
r! Ar i“1 2iπ j“1
rě1
which can also be rewritten:
Kn,c pxq “ exp
´ÿ 1
¯
Lbr
c,x rWr s
r!
rě1
where we introduced:
Lc,x f pxq “
n
ÿ
j“1
ˆ
(6.28)
xj
cj
(6.29)
8
As a consequence of Proposition 5.5, Wn P OpN 2´n q and has a 1{N expansion. Therefore, only a
finite number of terms contribute to each order in the n-kernels, and we find:
Proposition 6.5 Assume Hypothesis 1.3. Then, for any K ě ´1, we have the asymptotic expansion:
#
+
K
´ k`2
¯
ÿ
ÿ 1
Kn,c pxq “ exp
N ´k
Lx,c rWrtku s
,
(6.30)
r!
r“1
k“´1
where δ “ inf j dpxj , Aq is assumed larger than δN introduced in Definition 5.4. For a fixed K, it is
uniform for x in any compact of pCzAqn .
l
If ϕ : A Ñ C is a function such that ϕpsq
p P op|s|´r q when |s| Ñ 8, one could study by similar methods
řN
;A Hrϕs
the asymptotic expansion of the exponential statistics µVN,β
re
s where Hrϕs “ i“1 ϕpλi q, that we
would establish thanks to Corollary 6.2 up to opN ´Kprq q, where
Kprq “
Y r ´ p1 ` κ ` 2θq ]
2pκ ` θq
(6.31)
Note that Kprq ě 0 implies r ě 1 ` κ ` 2θ “ 7{2. In particular, we can deduce:
p
P
Proposition 6.6 The central limit theorem 5.6 holds for test functions ϕ such that |ϕpsq|
op|s|´p1`κ`2θq q when |s| Ñ 8.
40
7
Fixed filling fractions: 1{N expansion of the partition function
;A
In this Section, we restrict ourselves to the fixed filling fraction model, i.e. we study µVN,,β
for some
P Eg . Following § 6.2, it is again not difficult to consider potentials of the form V “ V1 ` V2 , with
V1 , V2 is holomorphic, so we will write down proofs only for holomorphic V .
7.1
Interpolation principle
Recall that, if pVt qt is a smooth family of potentials so that Bt Vt is holomorphic in a neighborhood of
A, we have:
˛
βN
dξ
Vt ;A
(7.1)
Bt Vt pξq W1Vt pξq.
Bt ln ZN,,β
“´
2 A 2iπ
We are going to interpolate in two steps between the initial potential V , and a potential for which the
partition function can be computed exactly by means of a Selberg β integral.
7.1.1
Reference potentials
We first describe a set of reference potentials. Let γ “ rγ ´ , γ ` s be a segment not reduced to a point,
and ρ˘ two elements of t˘1u. We introduce a probability measure supported on γ:
b
cγ,ρ
dσγ,ρ pxq “
px ´ γ ´ qρ´ pγ ` ´ xqρ` dx,
(7.2)
π
V
where the constant cΓ,ρ ensures that the total mass is 1. It is well-known that σγ,ρ “ µeqγ,ρ
following data:
;γ̃
for the
‚ if pρ´ , ρ` q “ p1, 1q, σγ,ρ is a semi-circle law, and it is the equilibrium measure for the Gaussian
potential
´
8
8
γ ´ ` γ ` ¯2
,
cγ,ρ “ `
Vγ,ρ “ `
(7.3)
x
´
´
2
pγ ´ γ q
2
pγ ´ γ ´ q2
on γ̃, any interval of γm “ R which is a neighborhood of γ.
‚ if pρ´ , ρ` q “ p´1, 1q, σγ,ρ is a Marčenko-Pastur law, and it is the equilibrium measure for a
linear potential:
4px ´ γ ´ q
2
Vγ,ρ “ `
,
cγ,ρ “ `
(7.4)
´
γ ´γ
γ ´ γ´
on γ̃, any interval of γm “ rγ ´ , `8r which is a neighborhood of γ.
‚ if pρ´ , ρ` q “ p1, ´1q, we have similarly a linear potential:
Vγ,ρ “
4pγ ` ´ xq
,
γ` ´ γ´
cγ,ρ “
2
γ` ´ γ´
(7.5)
on γ̃, any intevral of γm “s ´ 8, γ ` s which is a neighborhood of γ.
‚ if pρ´ , ρ` q “ p´1, ´1q, σγ,ρ is an arcsine law, and it is the equilibrium measure for a constant
potential:
Vγ,ρ “ 0,
cγ,ρ “ 1
(7.6)
on γ̃ “ γ “ γm .
41
V
;γ
γ,ρ m
of the initial model with such potentials are
When we choose γ̃ “ γm , the partition function ZN,β
special cases of Selberg β integrals, and therefore can be computed exactly. We have in general:
V
γ,ρ
ZN,β
;γm
´ “
‰ ´ γ ` ´ γ ´ ¯¯ ρ
“ exp ´ pβ{2qN 2 ` p1 ´ β{2qN ln
ZN,β ,
4
(7.7)
ρ
where ZN,β
do not depend on γ and their expression is given in Appendix B.2.
7.1.2
Step 1: interpolation with a reference potential
Ťg
Ťg
`
V ;A
Given A “ h“0 Ah “ h“0 ra´
h , ah s, we consider the support of the equilibrium measure µeq, , which
Ťg
Ťg
´
`
´ `
is of the form S “ h“0 Sh, “ h“0 rαh, , αh, s, with signs ρh “ pρh , ρh q indicating the soft of hard
nature of the edges. Let pUh q0ďhďg be a family of pairwise distinct neighborhoods of Ah . We denote:
Vref, pxq “
g
ÿ
ˆ
´
ÿ
1Uh pxq h VSh, ,ρh pxq `
2h1
h1 ‰h
h“0
Sh1 ,
By construction, Vref, is holomorphic in the neighborhood U “
σref, pxq “
g
ÿ
¯
ln |x ´ ξ|dσSh1 , ,ρh1 pξq .
Ťg
h“0
h σSh, ,ρh
(7.8)
Uh of A, and:
(7.9)
h“0
is the equilibrium measure for the potential Vref, on A. Indeed, it satisfies the characterization (1.20).
Notice that σref has same support as µeq, , edges of the same nature, and same filling fractions.
Besides, Vref satisfy the assumptions of 1.2. Then, if we consider the convex combination of potentials
Vs “ p1´sqV `sVref , it follows from the characterization (1.20) that the equilibrium measure associated
to Vs on A with filling fraction is precisely:
;A
s ;A
µVeq,
“ p1 ´ sqµVeq,
` s σref, .
(7.10)
;A
and σref, satisfy (5.2) with edges of the same nature, we conclude that if V
Besides, since both µVeq,
satisfies 1.2, so does the family pVs qsPr0,1s . Therefore, we can use Theorem 5.5 to deduce from (7.1)
the asymptotic expansion:
#
+
ˆ 1 ˛
V ;A
ZN,,β
β ÿ
dξ
tk`1u;Vs
´k
“ exp
N
ds
pVref pξq ´ V pξqq W1,
pξq .
(7.11)
Vref ;A
2 kě´2
ZN,,β
0
S 2iπ
7.1.3
Step 2: localizing the supports
We now have to analyze the partition function for the reference potential Vref defined by (7.8). When
g “ 0 (the one-cut regime), Vref coincides with one of the reference potentials, and we know that up
Vref ;A
to exponentially small correction, ZN,β
will be given by the Selberg integrals described case by case
in (B.6)-(B.8), so there is nothing more to do.
Assume now g ě 1. Let us define shortening flows on the support:
„ ´

´
`
`
α`
α´
α´
αh, `α`
´ `
h,
h, ´αh,
h, `αh,
h, `αh,
t
‚ if pρh , ρh q “ p1, 1q or p´1, ´1q, we set Sh, “
´
t,
`
t ,
2
2
2
2
“ ´
‰
`
`
`
´
t
‚ if pρ´
h , ρh q “ p´1, 1q, we set Sh, “ αh, , αh, ` tpαh, ´ αh, q ,
“ `
‰
`
`
´
`
t
‚ if pρ´
h , ρh q “ p1, ´1q, we set Sh, “ αh, ´ tpαh, ´ αh, q, αh, .
42
We consider the family of potentials on A:
g
ÿ
t
Vref,
pxq “
ˆ
´
ÿ
2h1
1Uh pxq h VSt,h ,ρh pxq ´
Sth1 ,
h1 ‰h
h“0
ln |x ´ ξ|dσSt
,h
¯
pξq
,
1 ,ρh1
(7.12)
Ťg
řg
for which the equilibrium measure is obviously h“0 h σSth, ,ρh and has support St “ h“0 Sth, .
t
Accordingly, Vref,
satisfies Hypothesis 5.1, uniformly for t in any compact of s0, 1s. Besides, the
Vt
ref,
partition function ZN,,β
0
α,h
;A
can be computed exactly in the limit t Ñ 0. If we introduce:
$
´
`
& pα,h ` α,h q{2
´
α
“
% ,h
`
α,h
we find that:
Vt
ref,
ZN,,β
lim
tÑ0
śg
`
pρ´
h , ρh q “ p1, 1q or p´1, ´1q
´ `
pρh , ρh q “ p´1, 1q
,
`
pρ´
h , ρh q “ p1, ´1q
if
if
if
(7.13)
;A
ˇ 0
ˇ 2
0 ˇN h h1
ˇα,h ´ α,h
,
1
ź
VSt ,ρ ;Tth
h h
“
h“0 ZN h ,β
(7.14)
0ďhăh1 ďg
where Tth is the maximum allowed interval associated to Sth which depends on the nature of the edges
(i.e. on ρh ) as described in § 7.1.1. We remark that the dependence in t factors and:
+
#
`
´ ¯
VSt ,ρ ;Tth
´ α,h
“
‰ ´ α,h
ρh
2 2
h h
ZN h ,β
ZN
(7.15)
“ exp ´ pβ{2qN h ` p1 ´ β{2qN h ln
h ,β ,
4t
ρh
where ZN
h is an analytic function of N h . The asymptotic expansion when N Ñ 8 of those factors
associated to reference potentials is described in Appendix B.2. We just mention that it is of the form:
´ ÿ
¯
1
γ,ρ
ZN,β
“ N pβ{2qN `γρ exp
N 2 Fβρ ,
(7.16)
kě´2
for γ “ rγ ´ , γ ` s. Therefore, we obtain:
Vref ;A
ZN,,β
+
#
g
g
”
´ÿ
¯
ı ´ α` ´ α´ ¯
ź
ˇ 0
,h
,h
ρ
0
β
2
2
h
ˇα,h ´ α,h1 |
“
ZN h ,β exp ´ pβ{2qN
h ` p1 ´ β{2qN ln
4
0ďhăh1 ďg
h“0
h“0
#
ˆ 1 ”
g
´ÿ
¯δ
ÿ
δk,´1
k,´2
ˆ exp
N ´k
ds pβ{2q
2h
` p1 ´ β{2q
s
s
0
kě´2
h“0
˛
ı*
s
dξ
tk`1u;Vref,
s
`
pBs Vref, qpξq W1,
pξq .
(7.17)
S 2iπ
ź
By construction, the integrand in the right-hand side is finite when s Ñ 0. The expression does
not make it obvious for the terms k “ ´2 and k “ ´1, but it can be checked explicitly since the
eigenvalues in different Ssh, decouple in the limit s Ñ 0, in the sense that:
Vs
W1 ref, pxq “
sÑ0
g
ÿ
VSs ,ρh
W1
h
pxq ` op1q,
(7.18)
h“0
V γ,ρ
and the expressions for the non decaying contributions to W1 ref when N is large are given in Appendix B.1.
43
7.2
Expansion of the partition function
ρ
We establish in Lemma B.1 that the partition functions for the reference potentials ZN,β
do have an
asymptotic expansion of the form:
´ ÿ
¯
ρ
N ´k Fβρ ` OpN ´8 q ,
ZN,β
“ N pβ{2qN `eρ exp
(7.19)
kě´2
where:
3 ` β{2 ` 2{β
β{2 ` 2{β
,
e`´ “ e´` “
,
12
6
Therefore, we have proved a part of Theorem 1.3:
1
γ``
“
e´´ “
´1 ` β{2 ` 2{β
.
4
Proposition 7.1 If pV, q satisfy Hypothesis 1.1 and 1.3 on A (instead of B), we have:
¯
´ ÿ
tku
V ;A
N ´k F,β ` OpN ´8 q .
ZN,,β
“ N pβ{2qN `e exp
(7.20)
(7.21)
kě´2
with a universal constant e “
řg
h“0 eρh
depending on β and the nature of the edges.
;A
Let ‹ the equilibrium filling fractions in the initial model µVN,β
. In order to finish the proof of
;A
Theorem 1.3, it remains to show that the stronger Hypotheses 1.2-1.3 for µVN,β
imply Hypothesis 5.1
V ;A
for the model µN,N ,β for the model with fixed filling fractions P Eg close enough to ‹ , that all
t´2u
coefficients of the expansion are smooth functions of , and that the Hessian of F
with respect
to filling fractions is negative definite. This last part is justified in Proposition A.3 proved in Appendix A.1, whereas to prove the first part, we rely on the basic result also proved in Lemma A.1 and
Corollary A.2 in Appendix A.1:
Lemma 7.2 If V satisfies Hypotheses 1.2-1.4, then pV, q satisfies Hypotheses 5.1 for P Eg close
t´1u
enough to ‹ . Besides, the soft edges αh‚ and W1, pxq are C 8 functions of , while the hard edges
remain unchanged, at least for close enough to ‹ .
t´1u
tku
‚
We observe that, once W1, and the edges of the support α,h
are known, the Wn, for any n ě 1
and k ě 0 are determined recursively by (5.51)-(5.49) and (5.70)-(5.73), where the linear operator
K´1 is given explicitly in (5.21)-(5.27), and thus depend also analytically on close enough to ‹ .
tku
Similarly, F for k ě 0 are obtained from (7.11)-(7.17), which shows their analyticity for close
enough to ‹ .
tku
tku
Corollary 7.3 If V satisfies Hypotheses 1.2-1.4, then Wn, and F
enough to ‹ .
are C 8 functions of P Eg close
l
This concludes the proofs of Theorem 1.3 and Corollary 1.7 announced in Section 1.4.
8
8.1
Asymptotic expansion in the initial model in the multi-cut
regime
The partition function
;A
We come back to the initial model µVN,β
, and we assume Hypotheses 1.2-1.4 with number of cuts
pg ` 1q ě 2. We remind the notation N “ pNh q1ďhďg for the number of eigenvalues in Ah , and the
řg
number of eigenvalue in A0 is N0 “ N ´ h“1 Nh . The Nh are here random variables, which take the
V ;A
V ;A
value N with probability ZN,,β
{ZN,β
. We denote ‹ the vector of equilibrium filling fractions, and
N‹ “ N ‹ . Let us summarize four essential points:
44
‚ We have established in Theorem 1.4 an expansion for the partition function with fixed filling
fractions:
´ ÿ
¯
N!
tku
V ;A
śg
ZN,,β
“ N pβ{2qN `e exp
N ´k F,β ,
(8.1)
h“0 pN h q!
kě´2
where e are independent of the filling fractions.
‚ By concentration of measures, we have established in Corollary 3.8 the existence of constants
C, C 1 ą 0 such that, for N large enough,
‰
1
2
;A “
µVN,β
|N ´ N‹ | ą ln N ď eCN ln N ´C N ln N .
(8.2)
tku
‚ Thanks to the strong offcriticality assumption, we have after Lemma 7.2 that F,β is smooth
when is in the vicinity of ‹ . From there we deduce that, for any K, k ě ´2, there exists a
constant Ck,K ą 0 such that:
ˇ
ˇ
tku
K´k
ˇ
ˇ
ÿ
pF‹,β qpjq
ˇ ´k tku
bj ˇ
´pk`jq
¨ pN ´ N‹ q ˇ ď Ck,K N ´pK`1q |N ´ N‹ |K´k`1 . (8.3)
N
ˇN FN {N,β ´
ˇ
ˇ
j!
j“0
t´2u
‚ We establish in Proposition A.3 in Appendix A.1 that the Hessian pF‹,β q2 is negative definite.
We now proceed with the proof of Theorem 1.5.
8.1.1
Taylor expansion around the equilibrium filling fraction
By the estimate (8.2), we can write:
V ;A
ZN,β
V ;A
ZN,
‹ ,β
V ;A
¯
´
ÿ
ZN,N
N!
N!
{N,β
V ;A
śg
ZN,N
p1 ` rN q, (8.4)
“
{N,β
V
;A
h“0 Nh ! ZN,‹ ,β
h“0 Nh !
0ďN1 ,¨¨¨ ,Ng ďN
ÿ
“
śg
0ďN1 ,¨¨¨ ,Ng ďN
|N |ďN
|N ´N‹ |ďln N
with:
rN ď pg ` 1qN e´
C1
2
N ln2 N
.
(8.5)
And, we have, for any K ě ´2:
ÿ
N!
V ;A
śg
ZN,N
{N,β
N
!
h
h“0
0ďN ,...,N ďN
1
r
|N‚ ´N‹ |ďln N
ÿ
exp
“
K K´k
´ ÿ
ÿ
tku
N ´pk`jq
k“´2 j“0
0ďN1 ,...,Nr ďN
|N ´N‹ |ďln N
¯
pF‹ qpjq
¨ pN ´ N‹ qbj ` N ´pK`1q RK .
j!
(8.6)
And, since N ´pK`1qRK ď 1 for N large enough:
|eN
´pK`1q
RK
´ 1|
ď 2|N ´pK`1q RK |
K
ÿ
1
ď N ´pK`1q
2Ck,K pln N qK´k`1 ď CK
N ´pK`1q pln N qK`3 .
(8.7)
k“´2
where we finally used (8.3). Notice that, since ‹ is the equilibrium filling fraction, we have pF t´2u q1‹ “
0, and therefore, for any K ě 0:
exp
K K´k
´ ÿ
ÿ
tku
N ´pk`jq
k“´2 j“1
b2
“ eiπτ‹,β ¨pN ´N‹ q
pF‹,β qpjq
j!
`2iπv‹,β ¨pN ´N‹ q
¨ pN ´ N‹ qbj
¯
(8.8)
K
´
¯
ÿ
tku
1`
N ´k T‹,β rN ´ N‹ s ` OpN ´pK`1q pln N qK`3 q ,
k“1
45
where we have introduced:
t´1u
pF‹,β q1
v‹,β “
2iπ
t´2u
,
τ‹ “
pF‹,β q2
2iπ
,
(8.9)
and for any vector X with g components:
tku
T,β rXs “
k
ÿ
1
r!
r“1
´â
r pF tmi u qp`i q ¯
ÿ
,β
`1 ,...,`r ě1
m1 ,...,mr ě´2
ř
r
i“1 `i `mi “k
i“1
`i !
ř
¨ X bp
i“1r `i q
.
(8.10)
Since the number of lattice points N satisfying |N ´ N‹ | ď ln N is a Opln N q, we can write:
V ;A
ZN,β
V ;A
ZN,‹,β
!
b2
ÿ
eiπτ‹,β ¨pN ´N‹ q
“
`2iπv‹,β ¨pN ´N‹ q
K
´
¯)
ÿ
tku
N ´k T‹,β rN ´ N‹ s
1`
N1 ,...,Ng PZg
|N ´N‹ |ďηN
k“1
`OpN ´pK`1q pln N qK`4 q.
(8.11)
where we have set ηN “ ln N .
8.1.2
Waiving the constraint on the sum over filling fractions
Now, we would like to extend the sum over the whole lattice Zg . Let us denote λ‹,β “
t´2u
min Sp p´F‹,β q2 ą 0. For any α ą 0 small enough, there exists a constant C 2 ą 0 so that:
ˇ
ˇ
ÿ
b2
ˇ
ˇ
eiπτ‹,β ¨pN ´N‹ q `2iπv‹,β ¨pN ´N‹ q pN ´ N‹ qbj ˇ
ˇ
N PZr
|N ´N‹ |ěηN
ď
2
ÿ
C2
e´λ‹,β p1´αqg|N ´N‹ | |N ´ N‹ |j
N PZg
|N ´N‹ |ěηN
ď
C2
ÿ
2
Volg pnq pn ` 1qj e´λ‹,β p1´αqg n ,
(8.12)
něηN
where Volg pnq “ p2n`1qg ´p2n´1qg ď g 2g ng´1 is the number of points in Zg so that n ď |N ´N‹ | ă
n ` 1. Therefore:
ˇ
ˇ
ÿ
b2
ˇ
ˇ
eiπτ‹,β ¨pN ´N‹ q `2iπv‹,β ¨pN ´N‹ q pN ´ N‹ qbj ˇ
ˇ
N PZr
|N ´N‹ |ěηN
ď
C2 p1 ` αqg 2g
´ ÿ
pn ` 1qg´1`j e´λ‹ p1´αqgnηN
¯
něηN
ď
C3,j e
2
´λ‹ p1´αqgηN
,
(8.13)
where C3,j is a constant depending on j. In other words, by unrestricting the sum in (8.11), we only
2
make an error of order Ope´C4 pln N q q, which is OpN ´8 q. Then, we remark that:

´ ∇ ¯bj „
ÿ
´N‹
v
iπpN ´N‹ q¨τ‹,β ¨pN ´N‹ q`2iπv‹,β ¨pN ´N‹ q
bj
e
pN ´ N‹ q “
ϑ
pv‹,β |τ‹,β q.
(8.14)
0
2iπ
N PZg
We have thus proved:
#
+ „

V ;A
K
ÿ
ZN,β
“
‰
´N‹
´k tku ∇v
“
N T‹,β
ϑ
pv‹,β |τ‹,β q ` OpN ´pK`1q pln N qK`4 q.
V ;A
0
2iπ
ZN,‹,β
k“0
(8.15)
The term appearing as a prefactor of N ´k is bounded when N Ñ 8. So, by pushing the expansion
one step further, the error OpN ´pK`1q pln N qK`4 q can be replaced by OpN ´pK`1q q. This concludes
the proof of Theorem 1.5.
46
8.2
Deviations of filling fractions from their mean value
We now describe the fluctuations of the number of eigenvalues in each segment. Let P “ pP0 , . . . , Pg q
be a vector of integers such that P ´N ‹,h P opN 1{3 q when N Ñ 8. The joint probability for h P v0, gw
to find Ph eigenvalues in the segment Ah is:
;A
µVN,β
rN “ P s “ śg
V ;A
ZN,P
{N,β
N!
h“0
Ph !
(8.16)
V ;A
ZN,β
We remind that the coefficients of the large N expansion of the numerator are smooth functions of
P {N . Therefore, we can perform a Taylor expansion in P {N close to ‹ , and we find that provided
P ´ N ‹ P opN 1{3 q, only the quadratic term of the Taylor expansion remains when N is large:
;A
µVN,β
rN “ P s

g
´ „
¯
ı
¯´1
”´ ÿ
´N‹
ϑ
pβ{2qpPh ´ N ‹,h q ln N
pv‹,β |τ‹,β q
exp
0
h“0
¯
´1
2
t´2u
t´1u
ˆ exp
pF‹,β q ¨ pP ´ N ‹ qb2 ` pF‹,β q1 ¨ pP ´ N ‹ q ` op1q
(8.17)
2
“
In other words, the random vector ∆N “ p∆N1 , . . . , ∆Ng q defined by:
g
ÿ
∆Nh “ Nh ´ N ‹,h `
t´2u
t´1u
1
rpF‹,β q2 s´1
h,h1 pF‹,β qh1
(8.18)
h1 “1
t´2u
converges in law to a random discrete Gaussian vector, with covariance rpF‹,β q2 s´1 . We observe
t´1u
that, when β “ 2, F‹,β
A
A.1
“ 0 so that N ´ N ‹ converges to a centered discrete Gaussian vector.
Elementary properties of the equilibrium measure with
fixed filling fractions
Smooth dependence
In this section, we prove Lemma 7.2, i.e. we establish under some assumptions that the equilibrium
measure µVeq, depends smoothly on the potential V and the filling fractions . Let V be at least C 2
and confining on A. We know that the support consists of a finite union of pairwise disjoint segments:
g
ď
SV “
´;V
`;V
SV,h “ rα,h
, α,h
s,
SV,h ,
´;V
`;V
α,h
ă α,h
.
(A.1)
h“0
Upon squeezing A, we can always assume that it is the disjoint union of pg ` 1q pairwise disjoint
Ťg
t´1u;V
segments A “ h“0 Ah , which are neighborhoods of SV,h in R. The Stieltjes transform W
of
this equilibrium measure satisfies:
@x P SV,h ,
@h P v0, gw,
t´1u;V
W1,
˛
and
@h P v0, gw,
Ah
t´1u;V
px ` i0q ` W1,
px ` i0q “ V 1 pxq,
dξ
t´1u;V
W
pξq “ h .
2iπ 1,
(A.2)
(A.3)
The general solution of (A.2) takes the form:
t´1u;V
W1,
pxq “
1 ´ PV pxq
`
2 σpxq
˛
47
A
V 1 pxq ´ V 1 pξq σpξq ¯
dξ ,
ξ´x
σpxq
(A.4)
where PV is a polynomial of degree g which should be determined by (A.3), and we recall the notation:
g b
ź
´;V
`;V
px ´ α,h
qpx ´ α,h
q.
σpxq “
(A.5)
h“0
This implies that the equilibrium measure has a density, which can be written in the form:
g
ź
dµV pxq “ dx SV pxq
‚
‚ ρh
|x ´ α,h
| .
(A.6)
h“0
for some ρ‚h “ ˘1, and SV pxq is a smooth function in a neighborhood of the support. This rewriting
assumes SV pαh‚ q ‰ 0 when ρ‚h “ ´1.
Let I “ v0, gw ˆ t˘1u, and H “ tI P I ρI “ ´1u. We assume strong off-criticality as defined in
Hypothesis 1.2:
Hypothesis A.1 The initial data V r0s and r0s is strongly off-critical, in the sense that, for any
I P IzH, SpαI q ‰ 0 and S 1 pαI q ‰ 0.
‚;V r0s
For any ph, ‚q P H, we consider the values αI fixed and equal to αr0s,h . We introduce the open set:
ˇ
!
)
ź
ˇ
U “ αP
R ˇ all αI , for I P I are pairwise distinct ,
(A.7)
IPIzH
and the map:
pF , T q : U ˆ Rg rXs ˆ C 2 pAq ˆ Rg`1 ÝÑ R2g`2´|H| ˆ Rg`1
(A.8)
defined by:
@I P H
FI rα, P, V, s
@h P v0, gw
Th rα, P, V, s
˛
σrαspξq dξ V 1 pαI q ´ V 1 pξq
P pαI q
b
,
`
2iπ
αI ´ x
‚ ´ α ‚1
A
α
1
JPIztIu
h
h
˛
dξ
“ ´h `
wrα, P, V spξq,
(A.9)
Ah 2iπ
“ś
where:
wrα, P, V spxq
σrαspxq
“
“
1 ´ P pxq
`
2 σrαspxq
ź?
x ´ αI .
˛
A
dξ V 1 pxq ´ V 1 pξq σrαspξq ¯
,
2iπ
x´ξ
σrαspxq
(A.10)
(A.11)
IPI
By construction, the data of the equilibrium measure µV satisfies:
pF , T qrαV , PV , V, s “ 0.
(A.12)
We would like to apply the implicit function theorem to show that:
Lemma A.1 If Hypothesis A.1 holds and V r0s is C r (resp. analytic) with r ě 2, αV and PV are
C r´1 (resp. analytic) functions of p, V q close enough to pr0s, V r0sq.
¸
The contour integrals A used in this Appendix A.1 can be rewritten as integrals over the segment
S. Thus, these manipulations do not assume that V is analytic in a neighborhood of A, and it does
makes sense to consider only V of class C r in the statement of Lemma A.1.
48
Proof. To achieve this we need to show that pdα , dP qpF, T q when evaluated at a point such that
pF , T qpα, P, V, q “ 0. We first compute:
BTh
rα, P, V, s “
BX h1
˛
1
Ah
dξ
ξh
.
2iπ 2σrαspξq
(A.13)
It is well-known that the g ˆ g submatrix of (A.13) consisting of indices h, h1 P v1, gw is invertible
when α P U, and on top of that, we find:
@h1 P v0, gw,
g
ÿ
BTh
rα, P, V, s
BX h1
h“0
˛
1
“
A
1
dξ
ξh
ξ h dξ
δh1 ,g
“ ´ Res
“
. (A.14)
ξÑ8 σrαspξq
2iπ σrαspξq
2
In particular, this expression does not vanish for h1 “ g, hence (A.13) is invertible. Then, we compute:
˛
BFI
σrαspξq dξ V 2 pαI qpξ ´ αI q ´ pV 1 pαI q ´ V 1 pξqq
rα, P, V s “ δI,J
BαJ
2iπ
pαI ´ ξq2
A
ÿ FI rα, P, V s ´ FJ rα, P, V s
`
2pαI ´ αJ q
JPI
˛
´
P 1 pαI q
σrαspξq dξ V 2 pαI qpξ ´ αI q ´ pV 1 pαI q ´ V 1 pξqq ¯
?
“ δI,J ś
`
2iπ
pαI ´ ξq2
A
JPIztIu αI ´ αJ
`
˘
?
“ δI,J lim x ´ αI Bx wrα, P, V spxq ,
(A.15)
xÑaI
V r0s
V r0s
and the latter does not vanish when evaluated at pαr0s , P , V r0s, r0sq thanks to Hypothesis A.1.
So, pdα , dP qpF , T q is invertible at this point. Besides, if V is C r for r ě 2, then dpF , T q is C r´2 .
Therefore, by the implicit function theorem, for p, V q close enough to pr0s, V r0sq, there is a unique
function C r´1 function pαV , PV q so that pF , T qpαV , PV , V, q “ 0. By uniqueness, it must correspond
l
to the data of µVeq, .
Corollary A.2 If Hypothesis 1.2 holds for pV r0s, r0sq with V r0s not necessarily real-analytic but C r
with r ě 2, they hold also for pV, q close enough to pV r0s, q and V r0s C r , and the density (A.6), once
multiplied by σpxq, is C r´1 for such data.
V r0s
t´1u;V
Indeed, (A.4) shows that W1,
pxq is a linear function of P
by σpxq, it is a smooth function of .
A.2
pxq{σpxq and hence, once multiplied
l
Hessian of the value of the energy functional
We are now in position to prove:
Proposition A.3 If Hypothesis 1.2 holds for pV r0s, r0sq with V r0s not necessarily real-analytic but
t´2u;V
at least C 2 , F
is C 2 for p, V q close enough to pr0s, V r0sq, and the g ˆ g matrix τV with purely
imaginary entries:
t´2u;V
1 B 2 F
@h, h1 P v1, gw,
pτV qh,h1 “
(A.16)
2iπ Bh Bh1
is such that Im τV ą 0.
t´2u
Proof. We are going to justify that F
is a concave function of in the domain:
!
Eg “ Ps0, 1rg`1 ,
g
ÿ
h“0
49
)
h “ 1 ,
(A.17)
řg
and within Hypothesis 1.2, which will imply the result. Let e a vector of Rg`1 such that h“0 eh “ 0
and e ‰ 0. For any P Eg , we define t “ ` te, which belongs to Eg for t small enough, and
µt “ µVeq,t . Its characterization as an equilibrium measure imply that :
ˆ
U t pxq “ V pxq ´ 2 ln |x ´ y|dµt pyq
(A.18)
is locally constant on the support of µt , and µt pAh q “ h ` teh . Let us denote Uht the value of U t pxq
when x P Sh .
t´2u;V
Let us denote Fe ptq “ Ft
. After a classical result (see e.g. [AG97, Joh98]):
¨
ˆ
”
ı
β
Fe ptq “
ln |x ´ y|dµt pxqdµt pyq ´ V pxqdµt pxq
(A.19)
2
It follows from Corollary A.2 that the density of µt is C 1 away from the edges, and its derivative is
integrable at the edges, thus define a measure that we denote ν t . It has same support as µt , and
´
we deduce from the characterization of µt that x ÞÑ ´2 ln |x ´ y|dν t pyq is locally constant on the
support of µt , and ν t rAh s “ eh . We deduce that Fe ptq is C 1 and:
ˆ ´ ˆ
g
¯
β ÿ t
β
t
t
1
2 ln |x ´ y|dµ pyq ´ V pxq dν pxq “ ´
U eh
(A.20)
Fe ptq “
2 A
2 h“0 h
eh does not depend on t, whereas t ÞÑ Uht is C 1 as one can deduce from the expression (A.18). Thus,
Fe is C 2 and:
g ˆ
ÿ
Fe2 ptq “ β
ln |xh ´ y|dν t pyq eh
(A.21)
h“0
Ah
where xh is any point in the interior of Ah . This can be rewritten:
g ¨
g
ÿ
ÿ
2
Fe ptq “ β
2 ln |x ´ y|dν t pxq1Ah pxqdν t pyq1Ah pyq “ β
Qrν t 1Ah , ν t 1Ah s
h“0
(A.22)
h“0
It is well-known property (see e.g. [AG97, Dei99]) that, for any signed measure ν with total mass 0,
Qrν, νs ě 0, with equality iff ν “ 0. Since we chose e ‰ 0, the vector of measures pν t 1Ah q0ďhďg is
not identically zero, hence Fe2 ptq ă 0. In other words, Fe is strictly concave for any direction e, hence
t´2u
F
is a strictly concave function of P Eg so that Hypothesis 1.2 holds.
l
B
B.1
Model Selberg integrals
Non decaying terms in correlators
Let us denote W1ρ,γm the first point correlator in the model with potential Vρ,γm described in § 7.1.1.
ř
tku;γm ,ρ
Let us denote ∆ “ pγ ` ´ γ ´ q{4. It admits a 1{N expansion: W1 ρ, γm “ kě´1 N ´k W1
.The
t´1u;γm ,ρ
,
expression for the equilibrium measure gives access to W1
a
x ´ px ´ γ ´ qpx ´ γ ` q
t´1u;γm ,``
,
pxq “
W1
2
d2∆
1 ´
x ´ γ` ¯
t´1u;γm ,´`
W1
pxq “
1´
,
(B.1)
2∆
x ´ γ´
d
1 ´
x ´ γ´ ¯
t´1u;γm ,`´
W1
pxq “
1´
,
(B.2)
2∆
x ´ γ`
t´1u;γm ,´´
W1
pxq
“
1
a
,
´
px ´ γ qpx ´ γ ` q
50
(B.3)
and then we deduce from (5.49):
t0u;γ ,``
W1 m
pxq
t0u;γm ,´`
W1
pxq
`
´
¯
x ´ γ `γ
2 ¯´
1
1´
2
a
,
“
1´
“
´
2
β
px ´ γ ´ qpx ´ γ ` q px ´ γ ´ qpx ´ γ ` q
´
2¯
∆
t0u;γ ,`´
“ W1 m
px “ ´ 1 ´
.
(B.4)
´
β px ´ γ qpx ´ γ ` q
t0u;γ ,´´
W1 m
pxq
t0u
Besides, we find from (5.51) that all these models share the same W2 :
´
t0u;γ ,ρ
W2 m px1 , x2 q
B.2
`
´
¯
x1 x2 ´ px1 ` x2 q γ `γ
` γ´γ`
1
2
2
a
´
1
`
.
“
β 2px1 ´ x2 q2
px1 ´ γ ´ qpx1 ´ γ ` qpx2 ´ γ ´ qpx2 ´ γ ` q
(B.5)
Exact formulas for partition function
Let us denote:
ˆ
ź
|λi ´ λj |β
N
ź
βN
2
V`` pλi q
λ2
2
``
ZN,β
“
´`
ZN,β
*
" ”
N
ź
β ¯ N ı ´ βN ¯
βN 2 ´
Γp1 ` jβ{2q
` 1´
ln
p2πqN {2
.
“ exp ´
4
2 2
2
Γp1 ` β{2q
j“1
ˆ
N
ź
ź
βN
V´` pλq “ λ
“
|λi ´ λj |β
e´ 2 V´` pλi q dλi ,
´´
ZN,β
" ”
* N
βN 2 ´
β ¯ ı ´ βN ¯ ź Γp1 ` jβ{2qΓp1 ` pj ´ 1qβ{2q
“ exp ´
` 1´
N ln
.
2
2
2
Γp1 ` β{2q
j“1
ˆ
N
ź
ź
“
|λi ´ λj |β
dλi
RN 1ďiăjďN
β`p2´βqN
V`` pλq “
(B.6)
i“1
r´2,2sN 1ďiăjďN
2
dλi ,
i“1
RN
` 1ďiăjďN
“ 2N
e´
(B.7)
i“1
`
˘2
N
ź
Γp1 ` pj ´ 1qβ{2q Γp1 ` jβ{2q
.
Γp2 ` pN ´ 2 ` jqβ{2qΓp1 ` β{2q
j“1
(B.8)
These are the values of the reference partition functions given in (B.6)-(B.8). To emphasize that they
can be defined for N not restricted to be an integer by analytic continuation, we introduce a function
related to the Barnes double Gamma function:
´dˇ
¯
ˇ
Γ2 pN ; b1 , b2 q “ exp
(B.9)
ˇ ζ2 ps ; b1 , b2 , xq ,
ds s“0
where:
ζ2 ps ; b1 , b2 , xq “
1
Γpsq
ˆ
8
0
e´tx ts´1 dt
.
p1 ´ e´b1 t qp1 ´ e´b2 t q
(B.10)
Its properties are reviewed in [Spr09], in particular it solves the functional equation:
Γ2 px ` b2 ; b1 , b2 q “
Γ2 pxq ?
1{2´x{b1
2π b1
,
Γpx{b1 q
Γ2 p1 ; b1 , b2 q “ 1.
(B.11)
ΓpN ` 1q
Γ2 pN ` 1 ; 2{β, 1q
(B.12)
We deduce from (B.11) the representation:
N
ź
Γp1 ` jβ{2q “ p2πqN {2 pβ{2qβN
j“1
51
2
{4`N p1{2`β{4q
Therefore, we can recast the Selberg integrals as:
´
“
‰¯
``
ZN,β
“ exp ´ pβ{4qN 2 ln N ` pβ{4 ´ 1{2qN ln N ` N pβ{2q lnpβ{2q ` lnp2πq ´ ln Γp1 ` β{2q
“
ΓpN ` 1q
Γ2 pN ` 1 ; 2{β, 1q
´
“
‰¯
exp ´ pβ{2qN 2 ln N ` pβ{2 ´ 1qN ln N ` N β lnpβ{2q ` lnp2πq ´ ln Γp1 ` β{2q
“
Γ2 pN ` 1q
Γp1 ` N β{2qΓ22 pN ` 1 ; 2{β, 1q
´
“
‰¯
exp βN 2 ln 2 ` N p2 ´ βq ln 2 ` p3β{2q lnpβ{2q ` lnp2πq ´ ln Γp1 ` β{2q
ˆ
`´
ZN,β
ˆ
´´
ZN,β
ˆ
B.3
Γp2{β ` N ´ 1qΓpN ´ 1q
Γ3 pN ` 1qΓ2 p2N ´ 1 ; 2{β, 1q
2
Γp2{β ` 2N ´ 1qΓp2N ´ 1q Γ p1 ` N β{2qΓ32 pN ` 1 ; 2{β, 1qΓ2 pN ´ 1 ; 2{β, 1q
Large N asymptotics of the partition function
We need the asymptotic expansion of Barnes double Gamma function [Spr09]:
1´
β¯
3 ` β{2 ` 2{β
βx2 ln x 3βx2
`
`
1`
px ln x ´ xq ´
ln x
ln Γ2 px ; 2{β, 1q “xÑ8 ´
4
8ÿ 2
2
12
´χ1 p0 ; b1 , b2 q `
pk ´ 1q! Ek pb1 , b2 q x´k ` Opx´8 q,
(B.13)
kě1
Ek pb1 , b2 q are the polynomials in two variables appearing as coefficients in the expansion:
ÿ
1
“
Ek pb1 , b2 q tk
´b
t
´b
t
1
2
p1 ´ e
qp1 ´ e
q tÑ0 kě´2
(B.14)
which are expressible in terms of Bernoulli numbers. χps ; b1 , b2 q is the analytic continuation to the
complex plane of the series defined for Re s ą 2:
ÿ
1
χps ; b1 , b2 q “
.
(B.15)
pm
b
`
m2 b2 qs
1
1
2
pm1 ,m2 qPN ztp0,0qu
For instance:
lnp2πq
` ζ 1 p´1q
2
We remind also the Stirling formula for the asymptotic expansion of the Gamma function:
ÿ Bk`1
ln x lnp2πq
ln Γpxq “ x ln x ´ x ´
`
`
x´k ,
xÑ8
2
2
kpk
`
1q
kě1
χ1 p0 ; 1, 1q “ ´
(B.16)
(B.17)
where Bk are the Bernoulli numbers. We deduce the asymptotic expansions:
Lemma B.1
``
ln ZN,β
“
`
˘
´p3β{8qN 2 ` pβ{2qN ln N ` ´ 1{2 ´ β{4 ` pβ{2q lnpβ{2q ` lnp2πq ´ ln Γp1 ` β{2q N
“
lnp2πq
3 ` β{2 ` 2{β
ln N ` χ1 p0 ; 2{β, 1q `
` op1q
12
2
`
˘
´p3β{4qN 2 ` pβ{2qN ln N ` ´ 1 ` pβ{2q lnpβ{2q ` lnp2πq ´ ln Γp1 ` β{2q N
“
β{2 ` 2{β
lnpβ{2q lnp2πq
` 2χ1 p0 ; 2{β, 1q ´
`
` op1q
(B.19)
6
2
2
`
˘
pβ{2qN ln N ` ´ β{2 ` pβ{2q lnpβ{2q ` pβ{2 ´ 1q lnp2q ` lnp2πq ´ ln Γp1 ` β{2q N
`
`´
ln ZN,β
(B.18)
`
´´
ln ZN,β
27 ´ 13p2{βq ´ 13pβ{2q
´1 ` 2{β ` β{2
ln N ` 3χ1 p0 ; 2{β, 1q `
lnp2q
4
12
lnp2πq
´ lnpβ{2q `
` op1q
2
`
52
(B.20)
where the op1q have an asymptotic expansion in powers of 1{N .
Acknowledgments
The work of G.B. is supported by Fonds Européen S16905 (UE7 - CONFRA) and the Fonds National
Suisse (200021-143434), and he would like to thank the ENS Lyon and the MIT for hospitality when
part of this work was conducted. This research was supported by ANR GranMa ANR-08-BLAN-031101 and Simons foundation.
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