Critical correlation functions for
the 4-dimensional n-component |ϕ|4 model
Alexandre Tomberg
joint work with R. Bauerschmidt and G. Slade
Warwick
May 29, 2014
Alexandre Tomberg
May 30, 2014
1 / 21
Outline
1. Model and results
Definition of the model
Main results
2. Overview of the RG method
Approximation by the free field
Progressive integration
3. Perturbation theory calculations
Flow equations
Sketch of the proof for n = 1
Alexandre Tomberg
May 30, 2014
2 / 21
Definition of the model
I
We fix a discrete torus Λ = ΛN = Z4 /LN Z4 .
I
For every x ∈ Λ, we consider n-component continuous spins ϕx ∈ Rn .
I
Given g > 0, ν ∈ R, let dϕx be the Lebesgue measure on Rn , we
define the |ϕ|4 probability measure as
P
1
ν
2 + g |ϕ|4
Y
−
ϕ
(−∆ϕ)
+
|ϕ|
1
x
x∈Λ
2 x
2
4
e
dϕx .
Z
x∈Λ
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Note that for n = 1, this is a continuous version of the Ising model.
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We use h · ig ,ν,N to denote the expectation with respect the above
measure. We are interested in critical correlation functions, in the
infinite volume limit h · ig ,ν = limN→∞ h · ig ,ν,N .
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We also write hF ; G i = hFG i − hF i hG i, both in finite and infinite
volume, for the correlation or truncated expectation of F , G .
Alexandre Tomberg
May 30, 2014
3 / 21
Critical νc
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We define the susceptibility as the limit
X
χ(g , ν, n) = lim
hϕ10 ϕ1x ig ,ν,N .
N→∞
x∈ΛN
Theorem (BBS 2014)
For g > 0 small enough, there exists νc = νc (g , n) < 0 and a constant
C = C (g , n) such that as ν ↓ νc ,
C
χ(g , ν, n) ∼
ν − νc
log
1
ν − νc
n+2
n+8
.
Also, νc (g , n) = −ag + O(g 2 ) with a = (n + 2)(−∆−1
) > 0 (the
Z4 0,0
Laplacian is the lattice Laplacian on Z4 , and its negative inverse is the
massless lattice Green function).
Alexandre Tomberg
May 30, 2014
4 / 21
Main result for n = 1
Theorem
Let n = 1 and g > 0 be sufficiently small. There exist constants
C1 , C10 > 0 such that as |a − b| → ∞,
C1
1
hϕa ; ϕb ig ,νc =
,
1+O
|a − b|2
log |a − b|
C10
1
log log |a − b|
2
2
hϕa ; ϕb ig ,νc =
1+O
.
|a − b|4 (log |a − b|) 23
log |a − b|
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This theorem was proven previously by K. Gawȩdzki and A. Kupiainen
using a different renormalisation group approach.
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A closely related version was also analysed by J. Feldman, J. Magnen,
V. Rivasseau, and R. Sénéor.
Alexandre Tomberg
May 30, 2014
5 / 21
Main result for n = 1
Theorem
Let n = 1 and g > 0 be sufficiently small. There exist constants
C1 , C10 > 0 such that as |a − b| → ∞,
C1
1
corrections,
with
O log |a−b|
|a − b|2
C10
1
log log |a−b|
∼
corrections.
2 with O
log |a−b|
4
|a − b| (log |a − b|) 3
hϕa ; ϕb ig ,νc ∼
hϕ2a ; ϕ2b ig ,νc
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This theorem was proven previously by K. Gawȩdzki and A. Kupiainen
using a different renormalisation group approach.
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A closely related version was also analysed by J. Feldman, J. Magnen,
V. Rivasseau, and R. Sénéor.
Alexandre Tomberg
May 30, 2014
6 / 21
Main result (n ≥ 2)
Theorem
Let n ≥ 2 and let g > 0 be sufficiently small, depending on n. As
|a − b| → ∞, there exist constants Cn , Cn0 > 0 such that for all i,
h(ϕia ) ; (ϕib )ig ,νc ∼
Cn
with
|a − b|2
O
1
log |a−b|
corrections.
For the correlation of squares, we require that i 6= j. Then
#
"
0
1
1
C
n
−
1
n
h(ϕia )2 ; (ϕib )2 ig ,νc ∼
+
,
4
n+2
n |a − b|4 (log |a − b|) n+8
(log |a − b|)2 n+8
"
#
1 Cn0
1
1
j 2
i 2
h(ϕa ) ; (ϕb ) ig ,νc ∼
+
,
−
4
n+2
n |a − b|4
(log |a − b|) n+8
(log |a − b|)2 n+8
both with O
log log |a−b|
log |a−b|
Alexandre Tomberg
corrections.
May 30, 2014
7 / 21
Main result (n ≥ 2)
Theorem
Let n ≥ 2 and let g > 0 be sufficiently small, depending on n. As
|a − b| → ∞, there exist constants Cn , Cn0 > 0 such that for all i,
h(ϕia ) ; (ϕib )ig ,νc ∼
Cn
with
|a − b|2
O
1
log |a−b|
corrections.
For the correlation of squares, we require that i 6= j. Then
#
"
0
1
1
C
n
−
1
n
h(ϕia )2 ; (ϕib )2 ig ,νc ∼
+
,
4
n+2
n |a − b|4 (log |a − b|) n+8
(log |a − b|)2 n+8
"
#
↓
1 Cn0
1
1
j 2
i 2
h(ϕa ) ; (ϕb ) ig ,νc ∼
+
,
−
4
n+2
n |a − b|4
(log |a − b|) n+8
(log |a − b|)2 n+8
both with O
log log |a−b|
log |a−b|
Alexandre Tomberg
corrections.
May 30, 2014
8 / 21
Remarks about negative correlations
(for the case n ≥ 2)
I
Note that (ϕia )2 is more highly correlated with (ϕib )2 than is |ϕa |2
with |ϕb |2 , due to cancellations with the negative correlations of
(ϕia )2 with (ϕbj )2 for i 6= j. More precisely,
h|ϕa |2 ; |ϕb |2 ig ,νc ∼ n
1
Cn0
n+2 .
4
|a − b| (log |a − b|)2 n+8
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In fact, we prove that h|ϕa |2 i < ∞, the field has a typical size.
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Making one component large must come at the cost of making
another one small. Thus locally, negative correlations between
different components are to be expected.
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Our results show that this effect persists over long distances at the
critical point.
Alexandre Tomberg
May 30, 2014
9 / 21
Continuous-time WSAW model as n = 0 case
I
Our result also covers the computation of the critical generating
function for the “watermelon” network of p mutually- and
self-avoiding walks, as the n = 0 case of the |ϕ|4 model.
a
I
p=2
b
a
b
a
b
p=6
The parameter p corresponds to the power p in h(ϕ1a )p ; (ϕ1b )p ig ,νc . In
the |ϕ|4 model, we only allow p = 1, 2, but for the WSAW model, we
prove a formula valid for all p ≥ 1, namely
(p)
Wa,b (g , νc (0)) ∼
I
p=4
C
1
.
2p
|a − b| (log |a − b|) 12 (p2)
This extends the work of R. Bauerschmidt, D. Brydges, and G. Slade
on the critical two-point function of the 4-dimensional weakly
self-avoiding walk.
Alexandre Tomberg
May 30, 2014
10 / 21
Approximation by the free field
I
We restrict our attention to the case n = 1, for simplicity.
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The |ϕ|4 measure on Λ = ΛN has the density
X 1
1 −Ug ,ν,1 (ϕ)
1
1
e
dϕ, Ug ,ν,z (ϕ) =
zϕx (−∆ϕ)x + νϕ2x + g ϕ4x .
Z
2
2
4
x∈Λ
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For a given z0 > −1, we split Ug ,ν,1 (ϕ) into a Gaussian part and a
perturbation
Ug ,ν,1 (ϕ) = U0,m2 ,1 ((1 + z0 )−1/2 ϕ) + Ug0 ,ν0 ,z0 ((1 + z0 )−1/2 ϕ).
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So that for C = (−∆ΛN + m2 )−1 ,
EC F (1 + z0 )1/2 ϕ e −Ug0 ,ν0 ,z0 (ϕ)
.
F (ϕ) g ,ν,N =
EC e −Ug0 ,ν0 ,z0 (ϕ)
Alexandre Tomberg
May 30, 2014
11 / 21
Observable fields
I
A standard approach to compute the correlation function is to
introduce observable fields and then differentiate.
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This leads us to define for σa , σb ∈ R,
X
V0 = U0 −
ϕpx (σa 1x=a + σb 1x=b ) .
x∈Λ
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Then we have for p = 1, 2
ϕpa
Alexandre Tomberg
;
ϕpb g ,ν,N
∂ ∂ log EC e −V0 .
= (1 + z0 )
∂σa ∂σb 0
p
May 30, 2014
12 / 21
Progressive integration
I
We begin by decomposing the covariance
∆Z4 + m2
−1
=
∞
X
Cj .
j=1
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Since each Cj is independent of the torus Λ by the finite range
property, (Cj )xy = 0 if |x − y | > 12 Lj , for each N, we have
−∆ΛN + m2
−1
=
N−1
X
Cj + CN,N .
j=1
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We now set (EC θF )(ϕ) = EζC F (ϕ + ζ) to obtain the formula
EC F = ECN,N ◦ ECN−1 θ ◦ · · · ◦ EC1 θF ,
with Z0 = e −V0 (Λ) and Zj+1 = ECj+1 θZj . Then ZN = EC Z0 .
Alexandre Tomberg
May 30, 2014
13 / 21
Cumulant expansion
I
If we can define Vj+1 in such a way so that ECj+1 θe −Vj ≈ e −Vj+1 , we
can iterate and rewrite the evolution Zj 7→ Zj+1 in terms of the much
simpler Vj 7→ Vj+1 . We use the cumulant expansion
1
−V
3
EC θe
= exp −EC θV + EC θ(V ; V ) + O(V ) .
2
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EC θ(A; B) = EC θ (AB) − (EC θA) (EC θB).
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For any polynomial P,
EC θP = e
LC
P=
∞
X
1 n
L
n! C
n=0
I
!
P,
LC =
∂ ∂
1 X
.
Cuv
2
∂ϕu ∂ϕv
u,v ∈Λ
Also EC P = e LC 0 P, where LC |0 is the LC operator with derivatives
taken at ϕ = 0.
Alexandre Tomberg
May 30, 2014
14 / 21
Approximating the flow
I
A natural candidate for Vj+1 thus comes from the cumulant expansion
I
1
Vj+1 ≈ ECj+1 θVj − ECj+1 θ(Vj ; Vj )
2
P
We maintain locality of V , that is V = x∈Λ Vx , using a projection
operator Loc, that we will not discuss here.
We are able to preserve the form of Vj , that is for all j and p = 1, 2,
I
Vj;x =
I
gj 4 νj 2 zj
ϕ + ϕx + ϕx (−∆ϕ)x − uj
4 x
2
2
− (λj ϕpx + tj ) (σa 1x=a + σb 1x=b ) − qj σa σb ,
where λ0 = 1, u0 = t0 = q0 = 0 and g0 , ν0 , z0 are obtained from g , ν.
Constant terms: since ZN = ECN,N ZN−1 , the last integration will set
all ϕ = 0 in ZN . Thus, ZN ≈ e uN +tN (σa +σb )+qN σa σb and
∂ 2 p
p
p
log ZN ≈ const · qN .
hϕa ; ϕb ig ,ν,N = (1 + z0 )
∂σa ∂σb 0
Alexandre Tomberg
May 30, 2014
15 / 21
Flow of coupling constants
I
We set wj =
Pj
i=1 Ci ,
βj = (n + 8)
X
x∈Λ
I
I
(wj+1 )20x
−
(wj )20x
and γ =
(
0
n+2
n+8
(p = 1)
(p = 2).
We also define the coalescence scale jab to be the smallest j such that
(Cj )ab 6= 0. By the finite range property, (Cj )ab = 0 if |a − b| > 12 Lj ,
so jab = blogL (2|a − b|)c.
Then the coefficients in Vj+1 are given by
gj+1 = gj − βj gj2 + . . .
(
λj (1 − γβj gj ) + . . . (j < jab )
λj+1 =
λjab + . . .
(j ≥ jab )
p
2
qj+1 = qj + p!λj (wj+1 )ab − (wj )pab + . . .
I
Providing the control of the non-perturbative part of V , that we
indicated by the (. . .), is a major challenge, but is not part of our
discussion here.
Alexandre Tomberg
May 30, 2014
16 / 21
Sketch of proof for q
I
Let us assume that Vj+1 ≈ ECj+1 θVj − 12 ECj+1 θ(Vj ; Vj ) and
Vj;x =
I
I
gj 4 νj 2 zj
ϕ + ϕx + ϕx (−∆ϕ)x − uj
4 x
2
2
− (λj ϕpx + tj ) (σa 1x=a + σb 1x=b ) − qj σa σb .
We want to reach out and grab the coefficient of the constant
monomial containing σa σb in Vj+1 , call it πab Vj+1 .
πab ECj+1 θ (Vj ) = −qj and πab 12 ECj+1 θ(Vj ; Vj ) = 22 ECj+1 θ(λj ϕpa ; λj ϕpb ).
p
X 1
(p!)2 2
1
LnCj+1 (ϕpa ϕpb ) =
λ (Cj+1 )pab .
πab ECj+1 θ(Vj ; Vj ) = πab λ2j
2
n!
p! j
n=1
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Therefore, we get −qj+1 = −qj − p!λ2j (Cj+1 )pab . By expanding the
p
definition of wj , we have (Cj+1 )pab = (wj+1 )ab − (wj )ab , while the
formula that I claimed to be true was
qj+1 = qj + p!λ2j (wj+1 )pab − (wj )pab .
Alexandre Tomberg
May 30, 2014
17 / 21
Analyzing the g flow
I
Let us begin by analyzing the recursion
gj+1 = gj − βj gj2 .
n+8
.
16π 2
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For m2 = 0, βj → β∞ = β̄ log L, where β̄ =
I
If we assume that βj = β∞ , we can solve the recursion and obtain a
formula for gj
gj ∼
1
g0
∼
as j → ∞.
1 + g0 β∞ j
β∞ j
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In fact, we prove that, as |a − b| → ∞,
1
log log |a − b|
1
gjab =
1+O
∼
.
log |a − b|
log |a − b|
β̄ log |a − b|
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Recall that jab = blogL (2|a − b|)c, so that β∞ jab ∼ β̄ log |a − b|.
Alexandre Tomberg
May 30, 2014
18 / 21
Analyzing the λ flow
I
I
I
The recursion defining λ depends on gj and the coalescence scale jab .
(
(
λj (1 − γβj gj ) (j < jab )
0
(p = 1)
λj+1 =
where γ = n+2
(p = 2).
λjab
(j ≥ jab ),
n+8
g
gj+1 γ
∼ 1 − γβj gj .
Since gj+1
∼
1
−
β
g
,
we
can
write
j
j
gj
j
Inserting this into the recursion, produces a simpler expression for the
leading terms in the flow of λ for both choices p = 1, 2
λj+1 = λj (1 − γβj gj ) ∼ λ0
j
Y
(1 − γβk gk ) ∼
k=0
I
gj+1
g0
γ
.
In particular, the limit limm2 ↓0 λ2jab exists and obeys, as |a − b| → ∞,
lim λ2jab ∼
m2 ↓0
Alexandre Tomberg
1
log |a − b|
2γ
.
May 30, 2014
19 / 21
Analyzing the q flow
I
As (Cj )ab = 0 for all j < jab , we can sum the equation for qj and
obtain a telescoping sum
qN ≈
N
X
p!λ2j (wj+1 )pab − (wj )pab = p!λ2jab (wN )pab .
j=jab
I
I
By definition, as N → ∞, qN → q∞ (m2 ) = p!λ2jab (−∆Z4 + m2 )−p
ab .
−p
2
2
The limit of (−∆Z4 + m )ab as m ↓ 0 is well-known
−1
lim (−∆Z4 + m2 )−1
ab = (−∆Z4 )ab ∼
m2 ↓0
I
1
.
|a − b|2
Finally, for νc = νc (g , 1) and as |a − b| → ∞,
1
hφa ; φb ig ,νc ∼
,
|a − b|2
2
1
φa ; φ2b g ,νc ∼
2 ,
|a − b|4 (log |a − b|) 3
since γ =
n+2
n+8
Alexandre Tomberg
= 13 .
May 30, 2014
20 / 21
Thank you.
Alexandre Tomberg
May 30, 2014
21 / 21