Central Limit Theorem for Ginzburg-Landau Processes by John

Central Limit Theorem

for Ginzburg-Landau Processes

by

John Sheriff

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Department of StatisticsUniversity of Toronto

Copyright c© 2011 by John Sheriff

Abstract

Central Limit Theorem

for Ginzburg-Landau Processes

John Sheriff

Doctor of Philosophy

Department of Statistics

University of Toronto

2011

The thesis considers the Ginzburg-Landau process on the lattice Zd whose potential

is a bounded perturbation of the Gaussian potential. For such processes the thesis

establishes the decay rate to equilibrium in the variance sense is Cgt−d/2 + o

(t−d/2

),

for any local function g that is bounded, mean zero, and having finite triple norm;

|||g||| =∑

x∈Zd

∥∥∂η(x)g∥∥∞ . The constant Cg is computed explicitly. This extends the

decay to equilibrium result of Janvresse, Landim, Quastel, and Yau [JLQY99] for zero-

range process, and the related result of Landim and Yau [LY03] for Ginzburg-Landau

processes.

The thesis also considers additive functionals∫ t

0g(ηs)ds of Ginzburg-Landau processes,

where g is a bounded, mean zero, local function having finite triple norm. A central limit

is proven for a−1(t)∫ t

0g(ηs)ds with a(t) =

√t in d ≥ 3, a(t) =

√t log t in d = 2, and

a(t) = t3/4 in d = 1 and an explicit form of the asymptotic variance in each case. Cor-

responding invariance principles are also obtained. Standard arguments of Kipnis and

Varadhan [KV86] are employed in the case d ≥ 3. Martingale methods together with L2

decay estimates for the semigroup associated with the process are employed to establish

the result in the cases d = 1 and d = 2. This extends similar results for noninteracting

random walks (see[CG84]), the symmetric simple exclusion processes (see [Kip87]), and

the zero-range process (see [QJS02]).

ii

Dedication

In loving memory of Molly

iii

Acknowledgements

There are many people without whom this thesis would not have been possible and I

cannot begin to thank them all. However, I would like to single out a few of those people

who played a critical role in this process. My supervisor Dr. Jeremy Quastel not only

suggested the problem studied within these pages, but his guidance, encouragement,

and patience were invaluable as I worked my way towards a solution. My committee

members Dr. Keith Knight and Dr. Jeffrey Rosenthal, and my external examiner Dr.

Sunder Sethuraman were generous with their time and made many important suggestions

towards improving the thesis.

My wife, Constance Sheriff, has been by my side for more than twenty years. Her

love and support have sustained me through good times and bad. She believed in me

even when I did not always believe in myself, and I am not sure that I will ever be able

to repay that faith. I would also like to thank my parents for everything that they have

made possible for me.

It was a pleasure sharing so much of my graduate experience with my friend Dr.

Hanna Jankowski. Our time together studying, talking, and laughing will always be

among my fondest memories.

I would like to extend my thanks to the faculty and staff of the Department of

Statistics. Your efforts produced a learning environment that was challenging, enriching,

and collegial. I am grateful for the time that I was able to spend in your midst and for

the role that you all played in my development.

I also wish to acknowledge the financial support provided by NSERC and the Uni-

versity of Toronto.

iv

Table of Contents

Table of Contents v

1 Introduction 1

2 Notation 6

3 Ginzburg-Landau Process 9

3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Results Related to Ginzburg-Landau Process . . . . . . . . . . . . . . . . 11

3.3 CLT and Decay to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 13

4 Approximation Results: Ginzburg-Landau 17

4.1 Finite Volume Approximation . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Mean Charge Approximation . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Relaxation to Equilibrium: Ginzburg-Landau 27

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Decay Rate for Specific Function . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Decay to equilibrium when g(ρ) = g′(ρ) = 0 . . . . . . . . . . . . . . . . 43

5.4 Entropy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.5 Proof of Lemma 5.3.4 and Lemma 5.3.5 . . . . . . . . . . . . . . . . . . . 70

v

5.6 Rate of decay to equilibrium for general functions . . . . . . . . . . . . . 77

6 Central Limit Theorem: Ginzburg-Landau 78

6.1 Outline of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Central Limit Theorem: Specific Case . . . . . . . . . . . . . . . . . . . . 84

6.3 CLT for general functions . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 Future Work 89

7.1 Central Limit Theorem for Ginzburg-Landau . . . . . . . . . . . . . . . . 89

7.2 Central Limit Theorem for ∇− φ Interface

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography 91

vi

Chapter 1

Introduction

The Central Limit Theorem (CLT) is among the most fundamental results in statistics.

While it appears in many forms, it is perhaps best known as stating that, subject to

some restrictions, the sum of independent, identically distributed random variables will

tend towards the same limiting distribution, that of a normal random variable.

Theorem 1.0.1. Let X1, X2, . . . be i.i.d with E[Xi] = 0, V ar(Xi) = σ2 < ∞, then

1√n

n∑i=1

Xi ⇒ N(0, σ2).

Building upon the earlier efforts of Bernoulli, de Moivre, Laplace, Poisson and Cheby-

shev, among others, Lyapunov is generally given credit for providing the first rigorous

proof of the central limit theorem with which we are now familiar [Ada74], [Hal98],

[LC86]. It is natural to ask to what extent these assumptions may be relaxed while still

maintaining the essential conclusion of the central limit theorem. While independence

was invariably an underlying assumption in the development of the central limit theo-

rem, the condition that the random variables be identically distributed was frequently

absent. For example, Poisson considered the limiting distribution of a weighted sum of

independent, but not identically distributed, random variables approximately 100 years

prior to the efforts of Lyapunov, Feller and Lindeberg in this area [LC86]. Of course

1

Chapter 1. Introduction 2

one may also relax the requirement that the variables be independent. Among the more

tractable situations to consider in this regard is dependence among random variables

that exists over a finite range. For example, one might consider a moving average pro-

cess from time series analysis, where the underlying series is composed of independent

random variables. In such settings, one may still establish a central limit theorem and,

in fact, the appropriate scaling is still√

n [HR48].

An important area of study where dependence among random variables need not be

of finite range is that of Markov processes in either discrete, or continuous time. A signif-

icant result in the area of Markov processes due to Kipnis and Varadhan, (see [KV86]),

states that given an ergodic, stationary, reversible Markov process Xt having invariant

distribution π, a mean zero function g, with g(X) ∈ L2(π) then,

1√t

∫ t

0

g(Xs)ds ⇒ N(0, σ2(g))

if σ2(g) < ∞, where

σ2(g) = limt→∞

1

tEπ

[(∫ t

0

g(Xs)ds

)2]

= 2

∫ ∞

0

Eπ[g(Xs)g(X0)]ds

In effect, when dependence is introduced among random variables, a central limit

theorem still exists, and under the traditional scaling, as long as the dependence among

variables decays quickly enough as a function of time. While general in nature, this

condition on σ2(g) is often difficult to check. Additionally, even in cases where it is

possible to calculate σ2(g), it need not be the case that σ2(g) is finite.

The realm of interacting particle systems brings us unavoidably in conflict with the

assumption of independence, and in even the simplest cases we find that σ2(g) is not finite.

For instance, consider the following example, as presented by Cox and Griffeath [CG84].

We are interested in a system of independent, symmetric nearest neighbour random

walks on Zd. Specifically, at each site x on lattice (Zd) place Nx ∼ Poisson(θ) particles.


At any time t, ηt(x) represents the number of particles at site x at time t. Each particle

waits an exponential amount of time and then jumps to a nearest neighbour.

Let g(ηt) = ηt(0)− θ. Then, Cox and Griffeath [CG84] were able to show

Eπ[g(ηt)g(η0)] = Eπ[(Pt/2g(η))2] = C(θ, d)t−d/2 + o(t−d/2),

where Pt is the semigroup associated with the reversible Markov process in this case. So,

by direct calculation, we see that σ2(g) is only finite in the case that d ≥ 3.

Hence, for d ≥ 3 an appeal to the result of Kipnis and Varadhan will allow us to

establish the existence of a central limit theorem, while in d = 1 and d = 2, σ2(g) is

unbounded and another approach is required. One might note at this point that random

walks are recurrent in dimensions one and two, but transient in higher dimensions. In

practice, Cox and Griffeath established their result through direct calculation in all di-

mensions, using cumulants. They established a central limit theorem using an alternate

scaling suggested by the preceding expectation. In particular:

1

a(t, d)

∫ t

0

g(ηs)ds ⇒ N(0, σ2),

where,

a(t, d) =

t3/4 d = 1√

t log t d = 2

t1/2 d ≥ 3

,

with 0 < σ2 < ∞ in each case.

There are a number of points to consider regarding this example which will bear upon

our subsequent efforts. The first is to note that since we are dealing with a system of

independent random walks, the necessary calculations in this case may be done explic-

itly. The second point of note, is that even this fundamental example of an interacting

particle system anticipates more complicated systems in the sense that we will encounter

essentially the same covariance structure. Thus, we should expect to invoke the result

of Kipnis and Varadhan in order to produce a central limit theorem in d ≥ 3, and we


should expect to employ essentially the same scalings as above in d = 1 and d = 2 as

we endeavour to establish central limit theorems for more complicated systems, where

interaction is present and explicit calculations are no longer possible. The final point

of note is that the central limit theorem here has been established for a quite specific

function. While this might suggest a likely candidate function for proving the central

limit theorem where interaction is present, one would ideally like to establish the central

limit theorem more generally.

Kipnis [Kip87] established the central limit theorem, again in the context of a specific

function of interest, for a model possessing interaction among particles. The model

considered was the symmetric simple exclusion model, wherein there is either 0, or 1

particle at any site on the d-dimensional lattice, Zd, and particles are restricted from

jumping to already occupied sites. The function for which the central limit theorem

was proven was precisely that which appeared in the example of Cox and Griffeath,

ηt(0) − θ, where θ represents the probability of observing a particle at any site in the

lattice under the equilibrium measure. Although the result may be generalized to a

broader class of functions [QJS02], the proof of the main result relies upon self duality

of the symmetric simple exclusion model which again makes explicit calculations possible

for this model. Unfortunately this property does not extend to other processes making

alternate approaches to the problem necessary.

Quastel, Jankowski, and Sheriff [QJS02] proved an analagous general result for the

symmetric zero range process, for any mean zero local function having polynomial bound.

In this model, multiple particles are permitted at any site on the lattice, with the jump

rate dependent upon the number of particles at a given site. Explicit calculations are

no longer possible and the result depends crucially on obtaining an explicit estimate of

the asymptotic variance up to leading order. Specifically, one must show that the L2

decay to equilibrium of the mean zero function g occurs at rate t−d2 , and one must be

able to explicitly compute the constant associated with the leading term. While such


decay to equilibrium is anticipated for a wide range of interacting particle systems, it

is currently only proven in the cases of symmetric simple exclusion and symmetric zero

range [JLQY99]. Naturally, this poses a significant obstacle to establishing the central

limit theorem result for a much larger class of systems.

The objectives of this thesis are to establish the L2 decay to equilibrium and to prove

the central limit theorem for the Ginzburg-Landau process. We will discuss the model

in greater detail in Chapter 3, but we note here that unlike symmetric simple exclusion

and zero range models which allow only a discrete number of particles at any site, the

Ginzburg-Landau process permits site values to be real numbers. This introduces both

benefits and challenges as we seek to establish the appropriate L2 decay rate and prove

the central limit theorem for the process.

In Chapter 2 we introduce some of the notation that will be used regularly in the

thesis. Chapter 3 discusses the Ginzburg-Landau model and some results related to the

model. As stated previously, the proof of the central limit theorem depends crucially

upon establishing the L2 decay to equilibrium of the mean zero function g occurs at

rate t−d2 . This is the subject of Chapter 4 through Chapter 5. With the appropriate

decay result in place, we take Chapter 6 to prove the central limit theorem for bounded

linear functions of the Ginzburg-Landau process. Finally, in Chapter 7 we briefly discuss

related avenues of inquiry.

Chapter 2

Notation

While we will introduce new notation as required throughout the thesis, we have chosen

to include a summary of some of the notation that will be encountered regularly.

Let ∆ represent the discrete Laplacian,

∆g(x) =∑y∼x

(g(y)− g(x)) , (2.0.1)

where y ∼ x indicates that y and x are nearest neighbours in Zd. The discrete Laplacian

will sometimes also be written in the equivalent form ∆d,

∆dg(x) =d∑

i=1

(g(x + ei) + g(x− ei)− 2g(x)) , (2.0.2)

where ei represents the unit vector in the ith direction.

Let ∇i represent the discrete gradient in the ith direction,

∇ig(x) = g(x + ei)− g(x),

where, as above, ei represents the unit vector in the ith direction, i = 1, . . . , d. The

discrete gradient is then given as

∇g(x) = (∇1g(x), . . . ,∇dg(x)) .

6

Chapter 2. Notation 7

For a positive integer L, denote by ΛL the cube centered at the origin, with side

length 2L + 1

ΛL = −L, . . . , Ld

Configurations in RZdwill typically be denoted by η. Configurations may be shifted

as τxη, where τxη(y) = η(x + y). Similarly, given a function g on RZd, τxg = g(τxη).

Averages will be represented as

Avgy∈ΛL

g(y) =1

|ΛL|∑y∈ΛL

g(y) (2.0.3)

When we are discussing averages of the process η over some cube Λn, we will let ηn

represent the average. That is ηn = 1|Λn|

∑y∈Λn

η(y).

In an effort to simplify notation, where desirable we will let G(·)u be the conditional

expectation of a local function u. Conditioning will typically arise in two forms, as follows

Gnu = E[u|ηn]

GΛnu = E[u|FΛn ],

where, for a positive integer L, FΛLis the σ-algebra generated by η(x), x ∈ ΛL. A

third form of conditioning will be introduced in Chapter 5.

Given a Markov process with generator L, and having invariant measure π, let S

represent the symmetric part of the generator. That is S =L+ L∗

2. For a function

g ∈ L2(π) define the H1 norm as

‖g‖21 = 〈g, (−S)g〉π,

where 〈·, ·〉π denotes expectation with respect to π. Let ‖ · ‖−1 denote the dual norm of

H1 with respect to L2(π). For g in L2(π) let

‖g‖2−1 = sup

h∈L2(π)

2〈g, h〉π − ‖g‖21. (2.0.4)

Chapter 2. Notation 8

Denote by H−1, the subset of L2(π), of all functions with finite ‖ · ‖−1 norm. Define the

triple norm of a function g to be,

|||g||| =∑

x∈Zd

∥∥∂η(x)g∥∥∞ , (2.0.5)

where ∂η(x)g =∂g

∂η(x)and ‖g‖∞ = sup

ηg(η)

Given a Markov process with generator L, a subset Ω of Zd, a cube Λ ⊂ Ω, a

probability measure ν on RΩ and a function g in L2(ν), define the Dirichlet form DΛ(ν, g)

of g on the cube Λ,

DΛ(ν, g) = −∫

gLΛgdν, (2.0.6)

where LΛ represents the generator restricted to Λ. When Λ = Zd, we will denote DΛ(ν, g)

by D(ν, g).

Chapter 3

Ginzburg-Landau Process

3.1 Model

For our purposes, the Ginzburg-Landau model is a system of interacting diffusions defined

on the d-dimensional lattice, although it may equally be defined in a continuous setting.

It may be considered as a microscopic model of magnetization. At each site x in the

d-dimensional lattice Zd, η(x) is a real valued random variable representing the spin, or

charge, at that site. Configurations of the state space RZdare denoted η. Spins at each site

evolve according to well defined dynamics that incorporate an explicit interaction among

neighbouring sites. The dynamics are given by an infinite series of linked differential

equations as

dηt(x) = ∆V ′(ηt(x))dt +d∑

i=1

∇idBt(x),

where Bt(x), x ∈ Zd are independent Brownian motions.

Given a local function g : RZd → R, we may also define the generator of the η-process

(Lg)(η) =1

2

∑x∼y

(∂η(x)−∂η(y))2g(η)−1

2

∑x∼y

(V ′(η(y))−V ′(η(x)))(∂η(y)−∂η(x))g(η), (3.1.1)

where V : R −→ R is some potential. We shall assume that

(A1) V (a) = 12a2 + F (a), where F : R −→ R is some smooth function such that

‖F‖∞ < ∞, ‖F ′‖∞ < ∞, ‖F ′′‖∞ < ∞

9

Chapter 3. Ginzburg-Landau Process 10

The assumption arises in the following context. We shall see that proving the required

decay estimate involves a precise estimate of the spectral gap for the process. This result

was proven by [LPY02] under assumption (A1) which we have therefore chosen to adopt.

In their paper, Landim, Panizo, and Yau [LPY02] assumed that the convex part of

potential to be Gaussian for simplicity, but speculate that the results would extend to

the case of a bounded perturbation of a convex potential.

Also, we require that the Ginzburg-Landau process reaches a state of finite entropy

in finite time, when started from a Dirac measure. Landim and Yau [LY03] proved such

a result and assumption (A1) will enter again in this context.

Denote by Z : R −→ R the partition function

Z(λ) =

∫ ∞

−∞eλa−V (a)da (3.1.2)

Let R : R → R be the density function ∂λ log Z(λ) = ∂ log Z(λ)/∂λ. It is smooth and

strictly increasing. Let Φ be the inverse of R so that

ρ =1

Z(Φ(ρ))

∫ ∞

−∞aeΦ(ρ)a−V (a)da , ρ ∈ R.

For λ in R, denote by νλ the product measure on RZddefined by

νλ(dη) =∏

x∈Zd

1

Z(λ)eλη(x)−V (η(x))dη(x)

Letting νρ = νΦ(ρ),

Eνρ [η(0)] =1

Z(Φ(ρ))

∫ ∞

−∞η(0)eΦ(ρ)η(0)−V (η(0))dη(0) = ρ,

and

Eνρ [V′(η(0))] =

1

Z(Φ(ρ))

∫ ∞

−∞V ′(η(0))eΦ(ρ)η(0)−V (η(0))dη(0) = Φ(ρ). (3.1.3)

For g ∈ L2(νρ), let g(ρ) = Eνρ [g] and define g′(ρ) = ∂αEνα [g]|α=ρ.

We note that the νρ are reversible for the Markov process with generator L. Finally, we


will denote the Dirichlet form associated to L as

D(νρ, h) = Eνρ [h(−L)h] =∑x∼y

Eνρ

[(∂h

∂η(x)− ∂h

∂η(y)

)2]

,

for some function h.

3.2 Results Related to Ginzburg-Landau Process

A series of papers (see [Fri87b], [Fri87a], [Fri89], [Fun89], [Fun90], and [GPV88]) explored

the hydrodynamic limit of the Ginzburg-Landau process in one and more dimensions and

in both the discrete and continuous cases. Equilibrium fluctuations for the process were

investigated by Spohn [Spo85]. These may be viewed respectively as a law of large

numbers result and related central limit theorem.

Landim, Yau, and Panizo [LPY02] studied the spectral gap and logarithmic Sobolev

inequalities for the Ginzburg-Landau process. We will make use of the following two

results which appear in their paper. The first of these is an equivalence of ensembles

result.

Lemma 3.2.1.

1. For ρ0 > 0, sup0≤ηk≤ρ0

|Gkg − g(ηk)| is bounded above by C‖g‖∞k−d, for some finite

constant C which depends only upon ρ0.

2. For ρ0 < 0, supρ0≤ηk≤0

|Gkg − g(ηk)| is bounded above by C‖g‖∞k−d, for some finite

constant C which depends only upon ρ0.

The second result of interest is the following spectral gap estimate.

Theorem 3.2.2. Let νΛl,M (·) = νρ

(·∣∣∑

x∈Λlη(x) = M

). Under the assumption (A1) on

the potential V , there exists a universal constant R0 ≥ 1 such that for all l ≥ 2, M ∈ R

EνΛl,M

[(g − EνΛl,M

[g])2

]≤ R0l

2DΛl(νΛl,M , g)

for all g ∈ L2(νΛl,M).


While Landim, Yau, and Panizo prove the spectral gap in one dimension, Kipnis and

Landim [KL99] outline a procedure for extending the procedure to higher dimension.

Kipnis and Landim [KL99] also include the following general perturbation result,

which will be of use in what follows.

Theorem 3.2.3. Assume that the generator L has a spectral gap of magnitude Γ−1:

Var(ν, f) ≤ ΓD(ν, f)

for every f in L2(ν). Let W be a mean-zero bounded function such that

〈W, (−L)−1W 〉ν < ∞,

where 〈W, (−L)−1W 〉ν = Eν [W (−L)−1W ]. Denote by λε the upper bound of the spectrum

L+ εW :

λε = supf :‖f‖2=1

〈f, (L+ εW )f〉ν = supf :‖f‖2=1

ε〈f,Wf〉ν −D(ν, f).

Then, for ε < (2 ‖W‖∞ Γ)−1

0 ≤ λε ≤ ε2

1− 2 ‖W‖∞ εΓ〈W, (−L)−1W 〉ν .

We conclude with an observation regarding the marginal density of the process. Let

gλ(x) = Z(λ)−1 exp λx− V (x), for some x ∈ Zd. Given our assumptions regarding

the potential V , the following lemma shows that gλ is bounded above and below by a

Gaussian density.

Lemma 3.2.4. Let gλ(x) = Z(λ)−1 exp λx− V (x). Then there exists a positive, finite

constant C1, depending only upon ‖F‖∞ such that

C−11

1√2π

e−(x−λ)2/2 ≤ gλ(x) ≤ C11√2π

e−(x−λ)2/2


Proof. By the definition of Z(λ),

gλ(x) =eλx−V (x)

∫eλx−V (x)dx

=e(x−λ)2/2−F (x)

∫e(x−λ)2/2−F (x)dx

,

after substituting for V and completing the square. Since −‖F‖∞ ≤ F (x) ≤ ‖F‖∞,

e(x−λ)2/2−F (x)

∫e(x−λ)2/2−F (x)dx

≤ e‖F‖∞

e−‖F‖∞e(x−λ)2/2

∫e(x−λ)2/2dx

= e2‖F‖∞ e(x−λ)2/2

√2π

Similarly,

e(x−λ)2/2−F (x)

∫e(x−λ)2/2−F (x)dx

≥ e−‖F‖∞

e‖F‖∞e(x−λ)2/2

∫e(x−λ)2/2dx

= e−2‖F‖∞ e(x−λ)2/2

√2π

.

This establishes the lemma with C1 = e2‖F‖∞ .

3.3 CLT and Decay to Equilibrium

We are actually interested in two related theorems, the central limit theorem and the

associated invariance principle. Our focus will be on the proof of the invariance principle

from which the central limit theorem is an easy consequence.

Theorem 3.3.1. Consider Ginzburg-Landau models as described previously. Fix an av-

erage charge ρ, and denote by Pρ the corresponding stationary process with marginals

νρ. Let g be a bounded, mean zero local function, with finite triple norm (2.0.5) , and

consider

Xt =1

a(t, d)

∫ t

0

g(ηs)ds, (3.3.1)

a(t, d) =

t3/4 d = 1√

t log t d = 2

t1/2 d ≥ 3


Under Pρ, Xt ⇒ X where X ∼ N(0, σ2) with

σ2 =

4Φ(ρ)

3√

π |Φ′(ρ)|3/2[g′(ρ)]2 d = 1

Φ(ρ)

2π |Φ′(ρ)|2 [g′(ρ)]2 d = 2

2 〈g, (−L)−1g〉 d ≥ 3

where χ(ρ) = Eνρ [(η0 − ρ)2], Φ(ρ) = Eνρ [V′(η(0))], and Φ′(ρ) = ∂αEνα [V ′(η(0))]|α=ρ.

Theorem 3.3.2. Consider Ginzburg-Landau models as described previously. Fix an av-

erage charge ρ, and denote by Pρ the corresponding stationary process with marginals νρ.

Let g be a bounded, mean zero local function, with finite triple norm, and consider

XNt =

1

a(N, d)

∫ Nt

0

g(ηs)ds, (3.3.2)

a(N, d) =

N3/4 d = 1√

N log N d = 2

N1/2 d ≥ 3

Under Pρ, XNt ⇒ Xt where Xt is

d = 1 : fractional Brownian motion,

cov.2Φ(ρ)

3√

π|Φ′(ρ)|3/2[g′(ρ)]2[t

3/22 + t

3/21 − |t2 − t1|3/2],

d = 2 : Brownian motion, covarianceΦ(ρ)

2π|Φ′(ρ)|2 [g′(ρ)]2 min(t1, t2),

d ≥ 3 : Brownian motion, covariance 2⟨g, (−L)−1g

⟩min(t1, t2)

where Φ(ρ) = Eνρ [V′(η(0))], and Φ′(ρ) = ∂αEνα [V ′(η(0))]|α=ρ.

The proof of the invariance principle will be the subject of Chapter 6. However, the

proof will depend crucially on establishing the appropriate rate of decay for mean zero

functions of the process in the variance sense, that is the L2 rate of decay. Specifically,

as suggested in the earlier examples involving independent random walks on a lattice

[CG84], symmetric simple exclusion [Kip87], and the zero range process [JLQY99], we


must establish that Eνρ [(Ptg(η))2] = Ct−d/2 + o(t−d/2), where g is a bounded, mean-zero,

local function having finite triple norm, C is some constant having the specific form

suggested in the theorem and Pt is the semigroup associated with the Ginzburg-Landau

process. Having established this rate of decay, we may immediately appeal to the result

of Kipnis and Varadhan [KV86] in order to demonstrate the central limit theorem and

invariance principle in d ≥ 3. Proof of the central limit theorem and invariance principle

in d = 1 and d = 2 will require additional arguments, which are subject of Chapter 6.

In Chapter 5 we prove the following theorem, which establishes just that rate of decay

and precisely specifies the constant C.

Theorem. 5.6.1 Let g be a bounded, mean zero, local function, with finite triple norm.

Then

Eνρ [(Ptg(η))2] =[g′(ρ)]2χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2) (3.3.3)


This result was shown previously, in the case of the symmetric zero-range model by

Janvresse, Landim, Quastel, and Yau [JLQY99], while Landim and Yau [LY03] estab-

lished, up to logarithmic correction, the O(t−d/2) rate of decay for the Ginzburg-Landau

model. While we will make use of important results and ideas developed in the Landim

and Yau result [LY03], we require a more precise estimate of the rate of decay if we are

to establish the central limit theorem. Therefore, while the approach detailed in Landim

and Yau is more direct and in some sense simpler, we will largely follow the approach

set out in Janvresse, Landim, Quastel, and Yau [JLQY99] in establishing the desired

result for the Ginzburg-Landau process. In particular, we will establish the desired rate

of decay for a quite specific function and then work to show that more general functions

of interest may be sufficiently well approximated by this special function. This differs

from the Landim and Yau result where the bound is more directly established for general

functions. Both results rely upon an entropy argument, but the change in models from


zero-range to Ginzburg-Landau necessitates some modifications in this argument. We

will address this modification when it arises in Section 5.3 and Section 5.4.

Chapter 4

Approximation Results:Ginzburg-Landau

Establishing the appropriate rate of decay will rely upon a series of approximations. The

first approximation that we will employ permits us to turn our attention from the process

in infinite volume to the process in finite volume.

4.1 Finite Volume Approximation

The finite volume approximation was presented in Janvresse, Landim, Quastel and Yau

[JLQY99] for the zero-range process. The following results, in the context of the zero-

range process, appear as Proposition 3.1, Lemma 6.1 and Lemma 6.2 respectively in their

paper.

Proposition 4.1.1. Fix γ > 0 and a local function h. Denote by sh the smallest integer

k such that the support of h is contained in the cube Λk. Also, let hs = Psh, where Ps is

the semigroup associated with the Ginzburg-Landau process, as introduced in the preceding

chapter. Then there exists a finite constant C(ρ) depending only on the average charge,

ρ, such that for each s ≥ max2, s2h and each L ≥ bγ√s log sc,

< (GΛLhs − hs)

2 >≤ C(ρ)

sγ< h2 >,

where 〈·〉 represents expectation with respect to νρ.

17

Chapter 4. Approximation Results: Ginzburg-Landau 18

We will return to the proof of this proposition after proving some preliminary results.

The generator L may be written as the sum of the generator over nearest neighbour

bonds b = (x, y).

(Lh) =∑

b

(Lbh),

where

(Lbh) = (∂η(x) − ∂η(y))2h(η)− (V ′(η(y))− V ′(η(x)))(∂η(y) − ∂η(x))h(η), (4.1.1)

A bond b = (x, y) is said to belong to ΛL if both x and y belong to ΛL. A bond

b = (x, y) is said to belong to ∂ΛL, the boundary of ΛL, if exactly one of x or y belongs

to ΛL. In such cases x will be identified as the end that belongs to ΛL.

Similarly, the Dirichlet form D(νρ, h) may be written as the sum of the Dirichlet form

over nearest neighbour bonds,

D(νρ, h) =∑

b

Db(νρ, h) =∑

b

Eνρ [h(−Lb)h]

Letting ht = Pth,

∂tGΛjht = ∂tEνρ [ht|FΛj

] = Eνρ [∂tht|FΛj] = Eνρ [Lht|FΛj

] = GΛjLht ,

for all j ≥ 1.

Hence,

∂t

⟨(GΛj

ht

)2⟩

= 2⟨GΛj

ht, GΛjLht

⟩

= 2Eνρ [Eνρ [ht|FΛj]Eνρ [Lht|FΛj

]]

= 2Eνρ [Eνρ [Eνρ [ht|FΛj]Lht|FΛj

]]

= 2Eνρ [Eνρ [ht|FΛj]Lht]

= 2⟨GΛj

ht,Lht

⟩


Lemma 4.1.2. There exists a finite constant C(ρ) such that for all β > 0,

⟨GΛj

h, (−L)h⟩ ≤

∑

b∈Λj

Db(νρ, h) +

(1 +

β

2

) ∑

b∈∂Λj

Db(νρ, h)

+C(ρ)

β

[⟨(GΛj+1

h)2

⟩−

⟨(GΛj

h)2

⟩]

Proof. Consider bonds that belong neither to Λj nor ∂Λj, the boundary of Λj. By the

definition of Lb,⟨GΛj

h,Lbh⟩

= 0 for such bonds.

Hence, one may write⟨GΛj

h, (−L)h⟩

as the sum of interior and boundary terms,

⟨GΛj

h, (−L)h⟩

=∑

b∈Λj

⟨GΛj

h, (−Lb)h⟩

+∑

b∈∂Λj

⟨GΛj

h, (−Lb)h⟩

(4.1.2)

Consider the first term in (4.1.2). For b ∈ Λj, since GΛjand Lb commute

⟨GΛj

h, (−Lb)h⟩

=⟨GΛj

h,GΛj(−Lb)h

⟩

=⟨GΛj

h, (−Lb)GΛjh⟩

= Db(νρ, GΛjh)

≤ Db(νρ, h) since the Dirichlet form is convex

Now consider the second term in (4.1.2). For b ∈ ∂Λj, GΛjand Lb no longer commute.

Instead, for general functions g and h,

〈g, (−Lb)h〉 = 〈∂bg, ∂bh〉 ,

where ∂bh(η) = ∂η(y)h(η)− ∂η(x)h(η).

Then (∂bGΛjh)(η) = ∂η(y)GΛj

h− ∂η(x)GΛjh = −∂η(x)GΛj

h, since y /∈ Λj.

Note, also that

GΛj∂bh = GΛj

(∂η(y) − ∂η(x))h

= −GΛj((Φ(ρ)− V ′(η(y)))h)− ∂η(x)GΛj

h.


Therefore, ∂bGΛjh = GΛj

∂bh + GΛj((Φ(ρ)− V ′(η(y)))h).

So,

⟨∂bh, ∂bGΛj

h⟩

=⟨∂bh, GΛj

∂bh⟩

+⟨∂bh,GΛj

((Φ(ρ)− V ′(η(y)))h)⟩.

Now consider the second term in (4.1.2). We have just shown that

∑

b∈∂Λj

⟨GΛj

h, (−Lb)h⟩

=∑

b∈∂Λj

〈∂bGΛjh, ∂bh〉

=∑

b∈∂Λj

〈∂bh,GΛj∂bh〉+

∑

b∈∂Λj

〈∂bh,GΛj((Φ(ρ)− V ′(η(y)))h)〉.

But, 〈∂bh,GΛj∂bh〉 = 〈GΛj

∂bh,GΛj∂bh〉 ≤ 〈∂bh, ∂bh〉 = Db(νρ, h) and employing the

Schwarz inequality,

〈∂bh,GΛj((Φ(ρ)− V ′(η(y)))h)〉

≤ β

2〈∂bh, ∂bh〉+

1

2β〈GΛj

((Φ(ρ)− V ′(η(y)))h), GΛj((Φ(ρ)− V ′(η(y)))h)〉

=β

2Db(νρ, h) +

1

2β〈(GΛj

((Φ(ρ)− V ′(η(y)))h))2〉,

for β > 0.

Therefore,

∑

b∈∂Λj

⟨GΛj

h, (−Lb)h⟩ ≤

(1 +

β

2

) ∑

b∈∂Λj

Db(νρ, u) (4.1.3)

+1

2β

∑

b∈∂Λj

⟨(GΛj

((Φ(ρ)− V ′(η(y)))h))2

⟩.

Now, consider a collection ψi, 1 ≤ i ≤ m of orthogonal vectors in a Hilbert space

H with inner product denoted by 〈〈·, ·〉〉. Then for every ψ in H,

m∑i=1

(〈〈ψ, ψi〉〉)2 ≤ max1≤i≤m

〈〈ψi, ψi〉〉〈〈ψ, ψ〉〉 (4.1.4)

For each bond b in ∂Λj, ((Φ(ρ)−V ′(η(y))) is FΛj+1 measurable and is mean zero with


respect to Eνρ

[·|FΛj

], so

GΛj((Φ(ρ)− V ′(η(y)))h) = Eνρ

[(Φ(ρ)− V ′(η(y)))h|FΛj

]

= Eνρ

[(GΛj+1

h)(Φ(ρ)− V ′(η(y)))|FΛj

]

= Eνρ

[(GΛj+1

h−GΛjh)(Φ(ρ)− V ′(η(y)))|FΛj

]

= GΛj

((GΛj+1

h−GΛjh)(Φ(ρ)− V ′(η(y)))

)

Since the functions (Φ(ρ)− V ′(η(y))), b ∈ ∂Λj are orthogonal with respect to

Eνρ

[·|FΛj

], then by inequalities (4.1.3), (4.1.4) and the preceding argument we have,

∑

b∈∂Λj

⟨GΛj

h, (−Lb)h⟩

≤(

1 +β

2

) ∑

b∈∂Λj

Db(νρ, h) +1

2β

∑

b∈∂Λj

⟨(GΛj

((Φ(ρ)− V ′(η(y)))h))2

⟩

≤(

1 +β

2

) ∑

b∈∂Λj

Db(νρ, h) +1

2βmaxb∈∂Λj

⟨(Φ(ρ)− V ′(η(y)))

2⟩⟨(

GΛj+1h−GΛj

h)2

⟩

≤(

1 +β

2

) ∑

b∈∂Λj

Db(νρ, h) +C(ρ)

β

(⟨(GΛj+1

h)2

⟩−

⟨(GΛj

h)2

⟩).

For positive integers k < K and β > 0, define U = Uk,K,β on L2(νρ) by

Uh = αk+1

⟨(GΛk

h)2⟩ +K−1∑

j=k

αj+1

⟨(GΛj+1

h−GΛjh)2

⟩+ αK+1

⟨(h−GΛK

h)2⟩ ,

where

αj = exp j/β.

Lemma 4.1.3. For each k, K, β ≥ 2, and t ≥ 0,

Uk,K,βht ≤ exp

C(ρ)t

β2

Uk,K,βh,

for some finite constant C(ρ).


Proof. We may write Uh as

Uh = αK+1

⟨h2

⟩−K∑

j=k+1

(αj+1 − αj)⟨(

GΛjh)2

⟩

Hence,

d

dtUht = −2αK+1D(νρ, ht)− 2

K∑

j=k+1

(αj+1 − αj)⟨GΛj

ht,Lht

⟩

Using Lemma 4.1.2, and noting that β ≥ 2

d

dtUht ≤ −2αK+1D(νρ, ht)

+ 2K∑

j=k+1

(αj+1 − αj)∑

b∈Λj

Db(νρ, ht)

+ 2βK∑

j=k+1

(αj+1 − αj)∑

b∈∂Λj

Db(νρ, ht)

+C(ρ)

β

K∑

j=k+1

(αj+1 − αj)∑

b∈∂Λj

(⟨(GΛj+1

h)2

⟩−

⟨(GΛj

h)2

⟩)

Since αj = exp j/β, we have that αj+1 ≥ β (αj+1 − αj). Therefore,

β

K∑

j=k+1

(αj+1 − αj)∑

b∈∂Λj

Db(νρ, ht) ≤K∑

j=k+1

αj+1

∑

b∈∂Λj

Db(νρ, ht).

A summation by parts shows that,

K∑

j=k+1

(αj+1 − αj)∑

b∈Λj

Db(νρ, ht)

= αK+1

∑

b∈ΛK+1

Db(νρ, ht)− αk+1

∑

b∈Λk+1

Db(νρ, ht)

−K∑

j=k+1

αj+1

∑

b∈Λj+1

Db(νρ, ht)−∑

b∈Λj

Db(νρ, ht)

≤ αK+1

∑

b∈ΛK+1

Db(νρ, ht)−K∑

j=k+1

αj+1

∑

b∈∂Λj

Db(νρ, ht).


Therefore,

−2αK+1D(νρ, ht) + 2K∑

j=k+1

(αj+1 − αj)∑

b∈Λj

Db(νρ, ht)

+2βK∑

j=k+1

(αj+1 − αj)∑

b∈∂Λj

Db(νρ, ht) ≤ 0.

So,

d

dtUht ≤ C(ρ)

β

K∑

j=k+1

(αj+1 − αj)∑

b∈∂Λj

(⟨(GΛj+1

h)2

⟩−

⟨(GΛj

h)2

⟩)

≤ C(ρ)

β2

K∑

j=k+1

αj+1

∑

b∈∂Λj

(⟨(GΛj+1

h)2

⟩−

⟨(GΛj

h)2

⟩)

≤ C(ρ)

β2Uht.

Finally, since

d

dtUht ≤ C(ρ)

β2Uht,

then

d

dslogUhs ≤ C(ρ)

β2.

Integrating from s = 0 to s = t and exponentiating,

Uk,K,βht ≤ exp

C(ρ)t

β2

Uk,K,βh

We are now in a position to prove the desired cutoff result.

Proof. (Proof of Proposition 4.1.1)

Fix a local function h and s ≥ max4, s2h. Let β =

√s, meaning β ≥ 2. Let k = b√sc,

K = bγ√s log sc, where bac represents the greatest integer less than, or equal to, a. This

means that the support of h is contained in Λk. Since the support of h is contained in


Λk, then referring to the definition of Uk,K,β,

K−1∑

j=k

αj+1

⟨(GΛj+1

h−GΛjh)2

⟩= 0,

αK+1

⟨(h−GΛK

h)2⟩ = 0.

Therefore, Uk,K,βh0 = αk+1 〈h2〉. Using Lemma 4.1.3 and the definition of Uk,K,βht,

αK+1

⟨(hs −GΛK

hs)2⟩ ≤ Uhs ≤ exp

C(ρ)s

β2

Uh0 = exp

C(ρ)s

β2

αk+1

⟨h2

⟩.

Since β =√

s,⟨(hs −GΛK

hs)2⟩ ≤ C(ρ)

αk+1

αK+1

⟨h2

⟩.

Also, as L ≥ K, then the definitions of α, k, K, and β imply that

⟨(GΛL

hs − hs)2⟩ ≤ ⟨

(GΛKhs − hs)

2⟩ ≤ C(ρ)

sγ

⟨h2

⟩,

for some finite constant C(ρ).

4.2 Mean Charge Approximation

Having restricted our attention to the finite volume problem via the previous approxima-

tion, we will now show that it is sufficient to consider average charge, by demonstrating

an appropriate bound on the difference between the function conditioned upon behaviour

within finite volume and the function conditioned upon mean charge within this region.

We wish to consider d-dimensional boxes, Λl and ΛL, where l < L. We will apply

spectral gap for the dynamics restricted to finite boxes in order to replace GΛLh by a

function that depends on the density of the charge in boxes of size Λl.

Assume l ≥ 2. Let R = (2l + 1)x, x ∈ Zd and consider an enumeration of this set:

R = x1, x2, . . . such that |xj| ≤ |xk| for j ≤ k. Let Ωj = xj + Λl and let Mj = Mj(η)

be the total charge in Ωj for the configuration η,

Mj =∑x∈Ωj

ηx.


Let q represent the total number of cubes with nonvoid intersection with ΛL and note

that q = O((L/l)d). For each j = 1, 2, . . . , denote by Mj the vector (M1, . . . , Mj).

Given a function h ∈ L2(νρ), Gl,Lh denotes the conditional expectation of h given Mq,

Gl,Lh = Eνρ [h|M1, . . . , Mq].

Now, as was the case in Janvresse, Landim, Quastel and Yau [JLQY99] (see Lemma

3.3), we wish to show that the spectral gap result may be used to bound the L2 distance

between GΛLh and Gl,Lh. First, note that if l and L are chosen such that (2L+1)/(2l+1)

is an odd number then

Gl,LGΛLh = Gl,Lh.

Since we may always increase L if necessary, without affecting our estimates, we may

assume that (2L + 1) is divisible by (2l + 1) and, in particular that (2L + 1)/(2l + 1) is

an odd number.

Lemma 4.2.1. For any h ∈ L2(νρ)

Eνρ

[(GΛL

h−Gl,LGΛLh)2

] ≤ R0l2DΛL

(νρ, h).

Proof. Since we may assume that (2L + 1)/(2l + 1) is an odd number

Eνρ

[(GΛL

h−Gl,LGΛLh)2

]= Eνρ

[(GΛL

h−Gl,Lh)2].

Now, fix a FΛL-measurable function h. For 1 ≤ j ≤ q, denote by Gj the decreasing

sequence of σ-algebras generated by M1, . . . , Mj and η(x), x ∈ Ωj+1

⋃. . .

⋃Ωq. Let h0 =

h and for 1 ≤ j ≤ q, let

hj = Eνρ [h|Gj].

Under this notation Gl,Lh = hq and also

Eνρ

[(h−Gl,Lh)2

]=

q−1∑j=0

Eνρ

[(hj+1 − hj)

2].


Now for 0 ≤ j ≤ q − 1 we may condition on Gj+1 and rely upon the definition of the

canonical measure νΛ,K in order to arrive at

Eνρ

[(hj+1 − hj)

2]

= Eνρ

[Eνρ

[(hj+1 − hj)

2|Gj+1

]]= Eνρ

[Var

(νΩj+1,Mj+1

, hj

)].

Given our result regarding the spectral gap, Theorem 3.2.2, and the fact that each Ωj is

a Λl-cube, we have that

Eνρ

[Var

(νΩj+1,Mj+1

, hj

)] ≤ R0l2Eνρ

[DΩj+1

(νΩj+1,Mj+1, hj)

].

Since the Dirichlet form is convex, the last expression is bounded above by

R0l2DΩj+1

(νρ, h). Summing over j concludes the proof of the lemma.

Chapter 5

Relaxation to Equilibrium:Ginzburg-Landau

5.1 Introduction

As indicated previously, relaxation, or decay, to equilibrium in the variance sense of

the Ginzburg-Landau process in infinite volume is a crucial component in establishing

the desired central limit theorem for the process. In finite volume, the rate of conver-

gence to equilibrium for a Markov process would typically involve estimating the spec-

tral gap of the generator of the process, together with spectral arguments to establish

that convergence to equilibrium takes place at an exponential rate. In the case of the

Ginzburg-Landau process in infinite volume, this approach is no longer possible since

the spectrum for conservative particle systems in infinite volume is continuous at zero.

Exponential rate of convergence is expected to be replaced by a polynomial rate of con-

vergence. Models such as independent random walks on the lattice [CG84], symmetric

simple exclusion [Kip87], and zero-range [JLQY99] suggest that the rate of decay to

equilibrium is O(t−d/2). With regard to the Ginzburg-Landau process, Landim and Yau

[LY03] demonstrated such a decay rate, up to a logarithmic correction. However, in order

to establish the desired central limit theorem, we require an explicit estimate of the decay

rate, particularly the leading order term.

When considering decay and central limit theorem results in the case of the simple

27

Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 28

exclusion process and the zero-range process, the actual form of the function, say g,

under consideration has limited impact upon the actual decay rate, entering only in the

form of a constant related to the mean value of the function. For example, in the case of

the zero-range process, decay to equilibrium occurs at rate C(ρ, d)[g′(ρ)]2t−d/2 + o(t−d/2),

where g′(ρ) = ∂αEνα [g]|α=ρ. It seems reasonable, therefore, to approach the problem of

proving the desired rate of decay, by proving the rate of decay first for some class of easy

functions and then extending the result to more general functions.

5.2 Decay Rate for Specific Function

Denote by Pt, t ≥ 0, the semigroup associated to the generator L introduced in (3.1.1)

and repeated below,

(Lg)(η) =1

2

∑x∼y

(∂η(x) − ∂η(y))2g(η)− 1

2

∑x∼y

(V ′(η(y))− V ′(η(x)))(∂η(y) − ∂η(x))g(η),

where V is some potential, satisfying assumption (A1) as before.

Ideally we might consider a function such as, η(0)−ρ, but we will need to modify this

slightly, taking our lead from the construction contained in Janvresse, Landim, Quastel

and Yau [JLQY99]. To that end, fix a smooth function

J : (−1, 1)d −→ R+ s.t.

∫J(u)du = 1,

with J(x) = 0 for x /∈ (−1, 1)d. For ε > 0, let K, k : R+ −→ N be two increasing, integer

valued functions defined by

K(t) = bt(1−ε)/2c, k(t) = bt2εc,

where, as before, bac stands for the greatest integer less than, or equal to, a. Then the

function we wish to consider is

Avgx∈ΛK

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ),


where Avg is as defined in (2.0.3). By definition of the function J , this is equivalent to

1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ).

Our first task is to show that such a function decays to equilibrium at the desired rate,

namely that,

Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2 =

χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2),


Before proceeding to the actual theorem and proof of this result we require a preliminary

construction.

Define the following symmetric random walk in continuous time,

pt(x, y) = e−2dt

∞∑n=0

(2dt)n

n!pn(x, y),

where pn(x, y) are the n-step transition probabilities for a simple symmetric random walk.

Let qt(x, y) = pΦ′(ρ)t(x, y) and define the function J(t, x) =∑

y∈Zd qt(x, y)J(y/K).

Then J(t, x) is the solution of

∂tJ(t, x) = Φ′(ρ)∆J(t, x), (5.2.1)

on R+ × Zd, with initial condition J(0, x) = J(x/K), where ∆ represents the discrete

Laplacian as introduced in (2.0.1).

We will ultimately demonstrate that the desired theorem relies upon establishing,

Eνρ

(1

|ΛK |∑

x∈Zd

J(t, x)τx Avgy∈Λk

(η(y)− ρ)

)2 =

χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2), (5.2.2)

and

Eνρ

[(Pt

[∫ t

0

1

|ΛK |∑

x∈Zd

(∆J)(t− r, x)τx Avgy∈Λk

(V ′(ηr(y))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ)) dr

])2]

≤ C(ρ)t−d/2−α, (5.2.3)

for some α > 0. Hence, we begin by establishing these two results.


Lemma 5.2.1.

Eνρ

(1

|ΛK |∑

x∈Zd


(η(y)− ρ)

)2 =

χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2),

where χ(ρ) = Eνρ [(η0 − ρ)2], Φ(ρ) = Eνρ [V′(η(0))], Φ′(ρ) = ∂αEνα [V ′(η(0))]|α=ρ, and

J(t, x) is defined as above.

Proof. We begin by writing Eνρ

(1

|ΛK |∑

x∈Zd


(η(y)− ρ)

)2 as

1

|ΛK |2∑

x,y∈Zd

J(t, x)J(t, y)Eνρ [τx(ηk − ρ)τy(ηk − ρ)] .

This latter expression is equal to

(1 + O

(k

K

))χ(ρ)

|ΛK |2∑

x∈Zd

J(t, x)2,

where, χ(ρ) = Eνρ [(η(0)− ρ)2]. Recall that J(t, x) =∑

y∈Zd qt(x, y)J(y/K). Therefore,

1

|ΛK |2∑

x∈Zd

J(t, x)2 =1

|ΛK |2∑

x∈Zd

∑

y∈Zd

qt(x, y)J(y/K)

2

.

This is bounded above by

(1 + A−1)1

|ΛK |2∑

x∈Zd

∑

y∈Zd

qt(x, 0)J(y/K)

2

+(1 + A)1

|ΛK |2∑

x∈Zd

∑

y∈Zd

(qt(x, y)− qt(x, 0))J(y/K)

2

,

for all A > 0. Since 1|ΛK |

∑y J(y/K) ≈ 1, this expression is equal to

(1 + A−1)∑

x∈Zd

qt(x, 0)2

+(1 + A)1

|ΛK |2∑

x∈Zd

∑

y∈Zd

(qt(x, y)− qt(x, 0))J(y/K)

2

.

(5.2.4)

By the definition of qt, and local central limit theorem

|qt(x, y)− qt(x, 0)| ≤ C(ρ, d)(|x| |y| /t)qt(x, 0),


where C(ρ, d) is some finite constant. Also,

∑

x∈Zd

qt(x, 0)2 = (8πΦ′(ρ)t)−d/2 + o(t−d/2).

Hence, the first term in (5.2.4) equals (8πΦ′(ρ)t)−d/2 plus lower order terms and the

second term in (5.2.4) is bounded above by C(ρ, d)t−d/2−α, for some finite constant C(ρ, d)

and α > 0, as desired.

We must now show that (5.2.3) holds which will depend upon the next two results.

The following proposition may be found in [KL99].

Proposition 5.2.2. For each function g in H−1 and t > 0,

Eπ

[(1√t

∫ t

0

g(Xs)ds

)2]≤ 20 ‖g‖2

−1 .

Lemma 5.2.3. Let ρ > 0, ρ0 > ρ and define the moment generating function M(θ) =

Eνρ

[eθη(x)

], for some θ ∈ R. Letting, I(θ, ρ0) = θρ0 − log M(θ),

P (ηk ≥ ρ0) ≤ e−|Λk|I(θ,ρ0),

where I(θ, ρ0) > 0, for small θ.

Proof. Since the marginal densities are bounded above and below by Gaussian densities,

the moment generating function is well defined. For θ > 0, and integer k > 0 Chebyshev’s

inequality implies

P (ηk ≥ ρ0) = P

(∑x∈Λk

η(x) ≥ |Λk|ρ0

)

≤Eνρ

[eθ

∑x∈Λk

η(x)]

eθ|Λk|ρ0

= e−|Λk|θρ0M(θ)|Λk|

= e−|Λk|(θρ0−log M(θ))

= e−|Λk|I(θ,ρ0)

(5.2.5)


It remains to show that I(θ, ρ0) > 0, for small θ. This result, in greater generality, may

be found in Durrett[Dur96].

Let κ(θ) = log M(θ). Since M(0) = 1, log M(0) = κ(0) = 0. Now,

θρ0 − κ(θ) =

∫ θ

0

ρ0 − κ′(u)du,

so by showing that κ′(θ) → ρ as θ → 0, we will have the desired result, namely I(θ, ρ0) >

0, for small θ. Once again, since the marginal densities are bounded above and below by

Gaussian densities, this moment generating function and κ(θ) are well defined and well

behaved at 0. But κ′(θ) = M ′(θ)/M(θ) and M ′(θ) → ρ as θ → 0 while M(θ) → 1 as

θ → 0.

Note that the preceding argument is easily altered to show that if ρ < 0, ρ0 < ρ,

P (ηk ≤ ρ0) ≤ e−|Λk|I(θ,ρ0),

where I(θ, ρ0) > 0.

We are now in a position to undertake the desired lemma.

Lemma 5.2.4.

Eνρ

[(Pt

[∫ t

0

1

|ΛK |∑

x∈Zd


(V ′(ηr(y))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ)

)dr

])2]

≤ Ct−d/2−α,

for some constant C, which depends upon ρ and d and α > 0,

Proof. The expectation is bounded above by

Eνρ

[([∫ t

0

1

|ΛK |∑

x∈Zd


(V ′(ηr(y))−Φ(ρ)−Φ′(ρ) (ηr(y)− ρ)

)dr

])2].


Recalling GkV′(η(0)) = E [V ′(η(0))|ηk], the previous equation is bounded above by

2Eνρ

(∫ t

0

1

|ΛK |∑

x∈Zd

(∆J) (t− r, x)τx Avgy∈Λk

(V ′(ηr(y))−GkV

′(ηr(0)))dr

)2 (5.2.6)

+ 2Eνρ

[(∫ t

0

1

|ΛK |∑

x∈Zd

(∆J) (t− r, x)

× τx Avgy∈Λk

(GkV

′(ηr(0))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ))dr

)2]. (5.2.7)

To simplify notation, let Uk(η) = Avgy∈Λk

(GkV

′(η(0))− Φ(ρ)− Φ′(ρ) (η(y)− ρ)). By Schwarz,

(5.2.7) is bounded above by

2t

∫ t

0

Eνρ

(1

|ΛK |∑

x∈Zd

(∆J) (t− r, x)τxUk(ηr)

)2 dr,

and this may be written as,

2t

∫ t

0

1

|ΛK |2∑

|x−y|≤2k

(∆J) (t− r, x) (∆J) (t− r, y)Eνρ [Uk(ηr)τx−yUk(ηr)] dr,

where this last step relies upon the fact that νρ is invariant and translation invariant,

that Uk(ηr) is mean zero, and that the cross products are independent for all x, y such

that |x− y| > 2k since νρ is a product measure and there is no overlap in the respective

Λk-cubes.

Noting that 2xy ≤ x2 +y2, and using the fact that νρ is translation invariant, we have

that

∑

|x−y|≤2k

(∆J) (t− r, x) (∆J) (t− r, y)Eνρ [Uk(η)τx−yUk(η)]

≤∑

|x−y|≤2k

1

2[(∆J) (t− r, x)]2 Eνρ

[(Uk(η))2

]

+1

2[(∆J) (t− r, y)]2 Eνρ

[(τx−yUk(η))2

]

= Eνρ

[(Uk(η))2

] ∑

|x−y|≤2k

1

2[(∆J) (t− r, x)]2 +

1

2[(∆J) (t− r, y)]2

≤ Eνρ

[(Uk(η))2

] ∑

x∈Zd

Ckd [(∆J) (t− r, x)]2 .


Hence (5.2.7) is bounded above by

Ctkd

|ΛK |2∫ t

0

Eνρ

[(Uk(η))2

] ∑

x∈Zd

[(∆J) (t− r, x)]2 dr

Now we proceed to show that

Eνρ

[(Uk(η))2

] ≤ C(ρ)k−2d.

Recall,

Uk(η) = Avgy∈Λk

[GkV′(η(0))− Φ(ρ)− Φ′(ρ) (η(y)− ρ)]

= GkV′(η(0))− Φ(ρ)− Φ′(ρ) (ηk − ρ) ,

where ηk = Avgy∈Λk

η(y). Thus,

|Uk(η)| ≤ |GkV′(η(0))|+ |Φ(ρ)|+ |Φ′(ρ)| |ηk − ρ| .

At this point, for convenience, we will make the assumption that ρ ≥ 0, but the same

argument may be used in the case that ρ < 0. Take ρ0 > ρ. Then

Eνρ

[U2

k

]= Eνρ

[U2

k I 0 ≤ |ηk| ≤ ρ0]+ Eνρ

[U2

k I |ηk| ≥ ρ0]. (5.2.8)

Considering the definition of Uk, the second term in (5.2.8) is bounded above by

4Eνρ

[(GkV

′(η(0)))2I |ηk| ≥ ρ0]

+ 4(Φ(ρ))2Eνρ [I |ηk| ≥ ρ0]

+ 4(Φ′(ρ))2Eνρ

[(ηk − ρ)2I |ηk| ≥ ρ0

].

Using the Schwarz inequality, this is bounded above by

4Eνρ

[(GkV

′(η(0)))4]1/2

Eνρ [I |ηk| ≥ ρ0]1/2

+ 4(Φ(ρ))2Eνρ [I |ηk| ≥ ρ0]

+ 4(Φ′(ρ))2Eνρ

[(ηk − ρ)4

]1/2Eνρ [I |ηk| ≥ ρ0]1/2 .


Recall (see Lemma 3.2.4) that the marginal density at any site is bounded above and

below by a Gaussian density. Hence, since V ′(x)2 ≤ C0(1 + x2) (by A1 for some fi-

nite constant C0, which depends upon ‖F ′‖∞), we have that Eνρ [(GkV′(η(0)))4] ≤

Eνρ [V ′(η(0))4] ≤ C1, and Eνρ [(ηk − ρ)4] ≤ C1, for some finite constant C1, depend-

ing upon ρ and ‖F‖∞, where V (x) = (1/2)x2 + F (x) (see A1). Therefore, the second

term in (5.2.8) is bounded above by C(ρ, F )Eνρ [I |ηk| ≥ ρ0]1/2, for some finite constant

C(ρ, F ). Since we have assumed that ρ > 0, Eνρ [I |ηk| ≥ ρ0] ≤ 2Eνρ [I ηk ≥ ρ0].By (5.2.5),

Eνρ [I ηk ≥ ρ0] = P (ηk ≥ ρ0) ≤ e−|Λk|I(θ,ρ0),

where I(θ, ρ0) = θρ0 − log M(θ). Therefore the second term in (5.2.8) is exponentially

small, which is clearly bounded above by C(ρ)k−2d, for some finite constant C(ρ).

Now consider the first term in (5.2.8). Since

Uk(η) = GkV′(η(0))− Φ(ρ)− Φ′(ρ) (ηk − ρ) ,

then

Eνρ

[Uk(η)2I0 ≤ |ηk| ≤ ρ0

]

≤2Eνρ

[(GkV

′(η(0))− Φ(ηk))2I0 ≤ |ηk| ≤ ρ0

]

+ 2Eνρ

[(Φ(ηk)− Φ(ρ)− Φ′(ρ) (ηk − ρ))

2I0 ≤ |ηk| ≤ ρ0

].

(5.2.9)

By the equivalence of ensembles result, Lemma 3.2.1, sup0≤ηk≤ρ0

|GkV′(η(0))− Φ(ηk)| and

sup−ρ0≤ηk≤0

|GkV′(η(0))− Φ(ηk)| are bounded above by C‖V ′‖∞k−d for some finite constant

C which depends only upon ρ0. Thus, the first term of (5.2.9) is bounded above by

C(V, ρ0)k−2d for some finite constant C(V, ρ0).

A Taylor series expansion reveals that, up to leading order, the second term of (5.2.9)

is bounded above by C(V, p)Eνρ [(ηk − ρ)4]. Since νρ is a product measure, this is bounded

above by C(V, p)k−2d as only products of the form (η(x)− ρ)2(η(y)− ρ)2, where x 6= y,

contribute to the leading order term.


Therefore, (5.2.7) is bounded above by

C(ρ)t

|ΛK |2k−d

∫ t

0

∑

x∈Zd

[(∆J) (t− r, x)]2 dr.

Since,

∂t∇J(t− r, x) = −Φ′(ρ)∆ (∇J(t− r, x)) (5.2.10)

this implies that

‖∇J(0, x)‖2 − ‖∇J(t, x)‖2 =

∫ t

0

∂r ‖∇J(t− r, x)‖2 dr

=

∫ t

0

∂r (∇J(t− r, x) ∇J(t− r, x)) dr

= 2Φ′(ρ)

∫ t

0

(∆J(t− r, x))2 dr.

Hence,∥∥∥∇J

( x

K

)∥∥∥2

≥ 2Φ′(ρ)

∫ t

0

(∆J(t− r, x))2 dr (5.2.11)

and,

2Φ′(ρ)

∫ t

0

1

|ΛK |∑

x∈Zd

[(∆J) (t− r, x)]2 dr ≤ 1

|ΛK |∑

x∈Zd

∥∥∥(∇J)( x

K

)∥∥∥2

.

As J was chosen to be a smooth function, a Taylor series expansion shows that, up to

leading order,

1

|ΛK |∑

x∈Zd

∥∥∥∇J( x

K

)∥∥∥2

=1

|ΛK |∑

x∈Zd

d∑i=1

1

K2

(∂iJ

( x

K

))2

,

where ∂iJ(

xK

)= ∂J

(xK

)/∂xi. Since J was chosen to be a smooth function that is also

compactly supported, this is of order K−2. Therefore, (5.2.7) is bounded above by

Ctkd

|ΛK | k−2dK−2 ≤ C(ρ, J)tk−dK−d−2.

By definition of K and k, (5.2.7) is bounded above by C(ρ, J)t−(d+ε)/2.

Now recall equation (5.2.6)

2Eνρ

(∫ t

0

1

|ΛK |∑

x∈Zd

(∆J) (t− r, x)τx

Avgy∈Λk

[V ′(ηr(y))−GkV

′(ηr(0))]

dr

)2 .


Integrating by parts,

∫ t

0

1

|ΛK |∑

x∈Zd

(∆J) (t− r, x)τx

Avgy∈Λk


′(ηr(0))]

dr

=1

|ΛK |∑

x∈Zd

(∆J) (0, x)

∫ t

0

τx

Avgy∈Λk


′(ηr(0))]

dr

+

∫ t

0

1

|ΛK |∑

x∈Zd

(∆∂rJ) (t− r, x)

∫ r

0

τx

Avgy∈Λk

[V ′(ηs(y))−GkV

′(ηs(0))]

ds dr.

Therefore, (5.2.6) is bounded above by

4Eνρ

(1

|ΛK |∑

x∈Zd

(∆J) (0, x)

∫ t

0

τx

Avgy∈Λk


′(ηr(0))]

dr

)2

+ 4Eνρ

(∫ t

0

1

|ΛK |∑

x∈Zd

(∆∂rJ) (t− r, x)

∫ r

0

τx

Avgy∈Λk

[V ′(ηs(y))−GkV

′(ηs(0))]

dsdr

)2 .

(5.2.12)

Now we may rewrite the first term in (5.2.12) as

4tEνρ

(1√t

∫ t

0

1

|ΛK |∑

x∈Zd

(∆J)( x

K

)τx

Avgy∈Λk


′(ηr(0))]

dr

)2 .

By taking into account the definition of ‖·‖−1, (see 2.0.4), we have,

4tEνρ

(1√t

∫ t

0

1

|ΛK |∑

x∈Zd

(∆J)( x

K

)τx

Avgy∈Λk


′(ηr(0))]

dr

)2

≤ Ct×

suph∈L2(νρ)

2

|ΛK |∑

x∈Zd

(∆J)( x

K

) ∫τx

Avgy∈Λk

[V ′(η(y))−GkV

′(η(0))]

hdνρ −D(νρ, h)

.

Since νρ is translation invariant,

Eνρ

[τx

Avgy∈Λk

[V ′(η(y))−GkV

′(η(0))]

h

]= Eνρ

[Avgy∈Λk

[V ′(η(y))−GkV

′(η(0))]

τ−xh

].

Let Wk(η) = Avgy∈Λk

[V ′(η(y))−GkV

′(η(0))]. Then Wk is FΛk

-measurable and

Eνρ [Wkτ−xh] = Eνρ

[Eνρ [Wkτ−xh|FΛk

]]

= Eνρ [WkGΛkτ−xh] .


Also, since D(νρ, h) is convex

D(νρ, h) ≥ 1

|Λk|∑

x∈Zd

DΛk(νρ, Gkτ−xh).

Thus the first term in (5.2.12) is bounded above by

Ct suph

2

|ΛK |∑

x∈Zd

(∆J)( x

K

) ∫τxWkhdνρ −D(νρ, h)

≤ Ct suph

2

|ΛK |∑

x∈Zd

(∆J)( x

K

) ∫WkGkτ−xhdνρ − 1

|Λk|∑

x∈Zd

DΛk(νρ, Gkτ−xh)

≤ Ct

|Λk|∑

x∈Zd

suph

∫2 |Λk||ΛK | (∆J)

( x

K

)WkGkτ−xhdνρ −DΛk

(νρ, Gkτ−xh)

≤ Ct|Λk||ΛK |2

∑

x∈Zd

[(∆J)

( x

K

)]2 ⟨Wk, (−LΛk

)−1 Wk

⟩.

By the spectral gap,⟨Wk, (−LΛk

)−1 Wk

⟩ ≤ k2 〈Wk,Wk〉 which, in turn is bounded above

by C(ρ)k2−d, since 〈Wk,Wk〉 is a variance term. Also,

1

|ΛK |∑

x∈ Zd

[(∆J)

(xK

)

1/K2

]2

−→∫

[(∆J)(x)]2 dx (5.2.13)

as K −→ ∞. Combining these results we see that the first term of (5.2.12) is bounded

above by C(J, ρ)tk2K−d−4, which by definition of k and K is bounded above by C(J, ρ)t−(d+1)/2.

For the second term of equation (5.2.12)

4Eνρ

(∫ t

0

1

|ΛK |∑

x∈Zd

(∆∂rJ) (t− r, x) dr

∫ r

0

τxWk (ηs) ds

)2

since J satisfies (5.2.1), this equals

4Eνρ

(∫ t

0

1

|ΛK |∑

x∈Zd

(∆Φ′(ρ)∆J) (t− r, x) dr

∫ r

0

τxWk (ηs) ds

)2 ,

which in turn equals

4Eνρ

(∫ t

0

1

|ΛK |∑

x∈Zd

Φ′(ρ)(∆2J

)(t− r, x) dr

∫ r

0

τxWk (ηs) ds

)2 .


Using the Schwarz inequality, this is bounded above by

4tΦ′(ρ)2Eνρ

∫ t

0

(1

|ΛK |∑

x∈Zd

(∆2J

)(t− r, x)

)2 (∫ r

0

τxWk (ηs) ds

)2

dr

which equals

4tΦ′(ρ)2

∫ t

0

Eνρ

r

(1√r

∫ r

0

1

|ΛK |∑

x∈Zd

(∆2J

)(t− r, x) τxWk (ηs) ds

)2 dr.

Since r ≤ t and by using Proposition 5.2.2, this term is bounded above by

4t2Φ′(ρ)2

∫ t

0

sup

h∈L2(νρ)

2

|ΛK |∑

x∈Zd

(∆2J

)(t− r, x)

∫τxWkhdνρ −D(νρ, h)

dr.

Now, repeating the previous argument, since Wk is FΛk-measurable and νρ is translation

invariant,

Eνρ [τxWkh] = Eνρ [WkGΛkτ−xh] .

Also, since D(νρ, h) is convex,

D(νρ, h) ≥ 1

|Λk|∑

x∈Zd

DΛk(νρ, Gkτ−xh).

Therefore,

4t2Φ′(ρ)2

∫ t

0

sup

h∈L2(νρ)

2

|ΛK |∑

x∈Zd

(∆2J

)(t− r, x)

∫τxWkhdνρ −D(νρ, h)

dr,

is bounded above by

Ct2Φ′(ρ)2

∫ t

0

|Λk||ΛK |2

∑

x∈Zd

[(∆2J

)(t− r, x)

]2 ⟨Wk, (−L)−1 Wk

⟩dr

≤ Ct2Φ′(ρ)2

∫ t

0

|Λk||ΛK |2

∑

x∈Zd

[(∆2J

)(t− r, x)

]2k2 〈Wk,Wk〉 dr

≤ C(ρ)t2Φ′(ρ)2

∫ t

0

k2

|ΛK |2∑

x∈Zd

[(∆2J

)(t− r, x)

]2dr.

Now, as before (see (5.2.10) and (5.2.11)), since

∂t∇∆J(t− r, x) = −Φ′(ρ)∆ (∇∆J(t− r, x))


this implies that

∥∥∥∇∆J( x

K

)∥∥∥2

≥ 2Φ′(ρ)

∫ t

0

(∆2J(t− r, x)

)2dr.

Hence, the second term of (5.2.12) is bounded above by

C(ρ)t2k2

|ΛK |2∑

x∈Zd

∥∥∥∇∆J( x

K

)∥∥∥2

.

As was the case with (5.2.13),

1

|ΛK |∑

x∈ Zd

[∥∥∇∆J(

xK

)∥∥1/K3

]2

−→∫‖∇∆J(x)‖2 dx,

as K −→∞.

Combining these results we see that the second term of (5.2.12) is bounded above by

C(J, ρ)tk2K−d−6, which by definition of k and K is bounded above by C(J, ρ)t−(d+1)/2.

We are now able to conclude this section with the desired theorem.

Theorem 5.2.5.

Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2 =

χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2), (5.2.14)

with Φ(ρ) and χ(ρ) as defined in (3.1.3) and Theorem 3.3 respectively.

Proof. Define the martingale Ms, 0 ≤ s ≤ t by

Ms =1∣∣ΛK(t)

∣∣∑

x∈Zd

J(t− s, x)τx Avgy∈Λk(t)

(ηs(y)− ρ)

−∫ s

0

(∂r + L)1∣∣ΛK(t)

∣∣∑

x∈Zd

J(t− r, x)τx Avgy∈Λk(t)

(ηr(y)− ρ)dr.

(5.2.15)

Since J is the solution of (5.2.1), the latter term of (5.2.15) is equal to

∫ s

0

(−Φ′(ρ))1

|ΛK |∑

x∈Zd


(ηr(y)− ρ) dr

+

∫ s

0

1

|ΛK |∑

x∈Zd

J(t− r, x)τx Avgy∈Λk

L (ηr(y)− ρ) dr.


A straightforward calculation shows that

L (ηr(y)− ρ) = ∆V ′(ηr(y)).

We further note that

1

|ΛK |∑

x∈Zd


L (ηr(y)− ρ)

=1

|ΛK |∑

x∈Zd


∆V ′(ηr(y))

=1

|ΛK |∑

x∈Zd


V ′(ηr(y)).

Also, by the definition of the discrete Laplacian,

∑

x∈Zd

(∆J) (t, x) = 0,

so we may add, or subtract∑

x∈Zd (∆J) (t, x)Φ(ρ) as desired. Therefore, the latter term

in (5.2.15) is equal to

∫ s

0

1

|ΛK |∑

x∈Zd


[V ′(ηr(y))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ)] dr

=

∫ s

0

1

|ΛK |∑

x∈Zd


W (ηr(y))dr,

where W (ηr(y)) = V ′(ηr(y))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ). Note that

M0 =1

|ΛK |∑

x∈Zd


(η(y)− ρ)

Then, by recalling the definition of our martingale Mt and by taking expectations(Eη[Mt] = Eνρ [Mt|η0] = M0

), we have that,

Eη[Mt] = Eη

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(ηt(y)− ρ)

]

− Eη

[∫ t

0

1

|ΛK |∑

x∈Zd


W (ηr(y))dr

]

=1

|ΛK |∑

x∈Zd


(η(y)− ρ).


Hence,

Eη

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(ηt(y)− ρ)

]

=1

|ΛK |∑

x∈Zd


(η(y)− ρ)

+ Eη

[∫ t

0

1

|ΛK |∑

x∈Zd


W (ηr(y))dr

](5.2.16)

and thus,

Eνρ

(Eη

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(ηt(y)− ρ)

])2

≤ (1 + A−1)Eνρ

(1

|ΛK |∑

x∈Zd


(η(y)− ρ)

)2

+ (1 + A)Eνρ

[(Eη

[∫ t

0

1

|ΛK |∑

x∈Zd


W (ηr(y))dr

])2]

(5.2.17)

for all A > 0.

By Lemma 5.2.1, the first term in (5.2.17) isχ(ρ)

[8πΦ′(ρ)t]d/2plus lower order terms. By

Lemma 5.2.4, the second term in (5.2.17) bounded above by C(ρ)t−d/2−α for some finite

constant C(ρ) and α > 0. Therefore,

limt→∞

td/2Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2

≤ (1 + A−1) limt→∞

td/2Eνρ

(1

|ΛK |∑

x∈Zd


(η(y)− ρ)

)2

for all A > 0. Letting A →∞, we have

limt→∞

td/2Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2

≤ limt→∞

td/2Eνρ

(1

|ΛK |∑

x∈Zd


(η(y)− ρ)

)2

=χ(ρ)

[8πΦ′(ρ)]d/2.


We may similarly rewrite (5.2.16) to show that

Eνρ

(1

|ΛK |∑

x∈Zd


(η(y)− ρ)

)2

≤ (1 + A−1)Eνρ

(Eη

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(ηt(y)− ρ)

])2

+ (1 + A)Eνρ

[(Eη

[∫ t

0

1

|ΛK |∑

x∈Zd


W (ηr(y))dr

])2]

for all A > 0. By repeating the preceding argument we have that,

χ(ρ)

[8πΦ′(ρ)]d/2≤ lim

t→∞td/2Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2 .

Therefore,

limt→∞

td/2Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2 =

χ(ρ)

[8πΦ′(ρ)]d/2,

thereby establishing that

Eνρ

(Pt

[1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

])2 =

χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2).

5.3 Decay to equilibrium when g(ρ) = g′(ρ) = 0

Our next step is to consider decay for some function g under the somewhat special

circumstance that g(ρ) = g′(ρ) = 0, and in particular demonstrate that

Eνρ [(Ptg(η))2] ≤ Ct−d/2−α,

for some finite constant C and α > 0.


As was the case in the previous section, the desired theorem relies upon establishing,

Eνρ

[(AvgΛK

[J

( y

K

)τy (Gkg)t

])2]≤ Ct−

d2−α, (5.3.1)

and

Eνρ

[(Pt

[∫ t

0

1

|ΛK |∑

x∈Zd


(V ′(ηr(y))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ)) dr

])2]

≤ C(ρ)t−d/2−α. (5.3.2)

for some constant C and α > 0. Hence, we begin by addressing these two results.

Proposition 5.3.1. For every bounded, local function g, with g(ρ) = g′(ρ) = 0, and

every smooth function J as defined above,

Eνρ

[(AvgΛK

[J

( y

K

)τy (Gkg)t

])2]≤ Ct−

d2−α,

for some constant C and α > 0.

Proof. Since Pt, t ≥ 0 is a contraction semigroup,

Eνρ

[(Pt Avg

ΛK

[J

( y

K

)τy(Gkg)

])2]

≤ Eνρ

[(AvgΛK

[J

( y

K

)τy(Gkg)

])2]

= Eνρ

[(AvgΛK

[J

( x

K

)τx(Gkg)

])(AvgΛK

[J

( y

K

)τy(Gkg)

])]

=1

|ΛK |2∑

|x−y|≤2k

J( x

K

)J

( y

K

)Eνρ [τx(Gkg)τy(Gkg)]

=1

|ΛK |2∑

|x−y|≤2k

J( x

K

)J

( y

K

)Eνρ [(Gkg)τx−y(Gkg)]

The final two equalities take advantage of the facts that νρ is a translation invariant

product measure and that Gkg(η) is a mean zero function which is FΛk-measurable.

Hence, the cross product is zero for all x, y ∈ ΛK such that |x− y| > 2k, as there is no

overlap in the respective Λk-cubes of interest.


In turn, this may be bounded above by

1

|ΛK |2∑

|x−y|≤2k

(J(x/K)2

2Eνρ

[(Gkg)2] +

J(y/K)2

2Eνρ

[(τx−yGkg)2]

).

Since νρ is a translation invariant measure, this equals

|Λ2k||ΛK |2

Eνρ

[(Gkg)2

] ∑x∈ΛK

J( x

K

)2

(5.3.3)

Now, since1

|ΛK |∑

x∈ΛK

J( x

K

)2

→∫

J(x, 0)2dx as K →∞, we have, for sufficiently large

K, that (5.3.3) is bounded above by,

C(J) |Λ2k||ΛK | Eνρ

[(Gkg)2

],

for some finite constant C(J). It remains to show that Eνρ [(Gkg)2] is at most O(k−2d).

Taking into account the definition of K and k, this means that

Eνρ

[(Pt AvgΛK

[J

(yK

)τy (Gkg)

])2]

is O(t−(1+ε)d/2), for ε > 0, which proves our proposi-

tion.

We will therefore show that Eνρ [(Gkg)2] is bounded above by C(g, ρ)k−2d for some

finite constant C(g, ρ). As before, for convenience, we will make the assumption that

ρ ≥ 0, but the same argument may be used in the case that ρ < 0. Take ρ0 > ρ, then

Eνρ

[(Gkg)2

]= Eνρ

[(Gkg)2I0 ≤ |ηk| ≤ ρ0

]+ Eνρ

[(Gkg)2I|ηk| ≥ ρ0

]. (5.3.4)

We will treat these terms separately, beginning with the second term in (5.3.4). Note

that the definition of Gkg indicates that |Gkg| ≤ ‖g‖∞. Therefore,

Eνρ

[(Gkg)2I|ηk| ≥ ρ0

] ≤ ‖g‖2∞Eνρ [I|ηk| ≥ ρ0] .

Since we have assumed that ρ > 0, Eνρ [I |ηk| ≥ ρ0] ≤ 2Eνρ [I ηk ≥ ρ0]. By (5.2.5),

Eνρ [I ηk ≥ ρ0] = P (ηk ≥ ρ0) ≤ e−|Λk|I(θ,ρ0),

where I(θ, ρ0) = θρ0−log M(θ). Therefore, ‖g‖2∞Eνρ [I|ηk| ≥ ρ0] is exponentially small,

which is clearly bounded above by C(ρ)k−2d, for some finite constant C(ρ).


Now consider the first term in (5.3.4). We have,

Eνρ

[(Gkg)2I0 ≤ |ηk| ≤ ρ0

] ≤2Eνρ

[(Gkg − g(ηk))

2 I0 ≤ |ηk| ≤ ρ0]

+ 2Eνρ

[(g(ηk))

2 I0 ≤ |ηk| ≤ ρ0].

(5.3.5)

An equivalence of ensembles argument, as before, may be employed to show that the first

term in (5.3.5) is bounded above by C(ρ)k−2d.

Finally, recalling that g(ρ) = g′(ρ) = 0, a Taylor series expansion reveals that, up

to leading order, the second term of (5.3.5) is bounded above by C(V, p)Eνρ [(ηk − ρ)4].

Since νρ is a product measure, this is bounded above by C(V, p)k−2d as only products of

the form (η(x)− ρ)2(η(y)− ρ)2, where x 6= y, contribute to the leading order term.

We also need to establish the following proposition.

Proposition 5.3.2. For every bounded local function g, with finite triple norm, such

that g(ρ) = g′(ρ) = 0, and every smooth function J as defined above,

Eνρ

[(gt − Avg

ΛK

[J

( y

K

)τy (Gkg)t

])2]≤ Ct−d/2−α,


However, the proof of this proposition is quite involved. As such, we will first indi-

cate how the proof of the desired theorem will unfold before undertaking the proof of

Proposition 5.3.2.

Theorem 5.3.3. Let g be a bounded, local function, with finite triple norm, such that

g(ρ) = g′(ρ) = 0. Then

Eνρ [(Ptg(η))2] ≤ Ct−d/2−α, (5.3.6)


Proof. For t ≥ 0, let gt stand for Ptg. Thus, gt is the solution of the backward equation

∂tgt = Lgt


g0 = g

Fix two constants, t0 > 0, R0 > 1. More explicit bounds on t0 and R0, will be developed

during the course of the proof, but we begin by requiring that t0 is sufficiently large that

the support of g is contained in Λbt0c.

For n ≥ 1, let tn = Rn0 t0. Set tn(t)+1 = t, where n(t) is the largest integer n such that

tn(t) ≤ t. As before, fix a smooth function with compact support

J : (−1, 1)d −→ R+ s.t.

∫J(u)du = 1

For ε > 0 small, let K, k : R+ −→ N be two increasing, integer valued functions defined

by

K(t) = bt(1−ε)/2n c, k(t) = bt2ε

n c,

for t ∈ [tn, tn+1). For each t ≥ 0

Eνρ [g2t ] ≤ 2Eνρ

[(gt − Avg

y∈ΛK

[J

( y

K

)τy (Gkg)t

])2]

+2Eνρ

[(Avgy∈ΛK

[J

( y

K

)τy (Gkg)t

])2]

.

By Proposition 5.3.1

Eνρ

[(AvgΛK

[J

( y

K

)τy (Gkg)t

])2]≤ Ct−d/2−α,

for some constant C and α > 0. By Proposition 5.3.2

Eνρ

[(gt − Avg

ΛK

[J

( y

K

)τy (Gkg)t

])2]≤ Ct−d/2−α,


We now turn our attention to the proof of Proposition 5.3.2.

Proof of Proposition 5.3.2.

Recall, that for j ∈ N, tj = Rj0t0, where R0, t0 are constants that will be more explicitly


defined in what follows. Also, n(t) is the largest integer n such that tn(t) ≤ t, where, for

simplicity we are assuming that tn(t)+1 = t. Finally, in what follows, unless otherwise

indicated, < · > represents expectation with respect to νρ.

Let vt = gt − 1

|ΛK |∑

y∈ΛK

J( y

K

)τy (Gkg)t. Then

t(d+2)/2 < v2t > −t

(d+2)/20 < v2

t0> =

∫ t

to

∂ss(d+2)/2 < v2

s > ds

=

n(t)∑j=0

∫ tj+1

tj

∂ss(d+2)/2 < v2

s > ds

Since K and k are constant on the intervals [tj, tj+1],

∂ss(d+2)/2 < v2

s >= −2s(d+2)/2D(νρ, vs) +

(d + 2

2

)sd/2Eνρ [v

2s ]

on such intervals. Thus,

t(d+2)/2 < v2t > −t

(d+2)/20 < v2

t0>= −2

∫ t

to

s(d+2)/2D(νρ, vs) ds+

(d + 2

2

) ∫ t

to

sd/2Eνρ [v2s ] ds

(5.3.7)

Therefore,

t(d+2)/2 < v2t > = −2

∫ t

to

s(d+2)/2D(νρ, vs) ds

+

(d + 2

2

) ∫ t

to

sd/2Eνρ [v2s ] ds + t

(d+2)/20 < v2

t0>,

and our objective is to demonstrate that,

−2

∫ t

to

s(d+2)/2D(νρ, vs) ds +

(d + 2

2

) ∫ t

to

sd/2Eνρ [v2s ] ds ≤ t1−α,

for some α > 0, which will therefore establish that

Eνρ [(Ptv)2] = Eνρ

[(gt − Avg

ΛK

[J

( y

K

)τy (Gkg)t

])2]≤ Ct−

d2−α.

Consider a trajectory X : R+ −→ Zd on the lattice such that X(s) is constant in the

intervals [tn, tn+1] and |X(tn)| ≤ (1/4)√

tn. Since the dynamics is translation invariant,


we may replace vs by τX(s)vs in the previous formula. Hence, after this substitution

(5.3.7) becomes

−2

∫ t

to

s(d+2)/2D(νρ, τX(s)vs)ds +

(d + 2

2

) ∫ t

to

sd/2Eνρ [(τX(s)vs)2]ds. (5.3.8)

We now wish to employ the finite volume approximation result, Proposition 4.1.1, to

replace the second term in (5.3.8) with one that is restricted to finite volume.

Step 1. Cutoff

For every L ≥ 1, the second term in (5.3.8) is equal to(

d + 2

2

) ∫ t

to

sd/2Eνρ

[(τX(s)vs −GΛL

τX(s)vs)2]ds

+

(d + 2

2

) ∫ t

to

sd/2Eνρ [(GΛLτX(s)vs)

2]ds

(5.3.9)

Now, by Proposition 4.1.1

Eνρ [(GΛLτX(s)vs − τX(s)vs)

2] ≤ C(ρ)

sγEνρ [(τX(s)v)2],

for some finite constant C(ρ), γ > 0, s ≥ max 2, s2τX(s)v

and L ≥ bγ√s log sc, where

sτX(s)v is the smallest integer k such that the support of τX(s)v is contained in the cube

Λk.

Consider an interval [tn, tn+1], and define L(s) = bγ√tn+1 log tn+1c on the interval

[tn, tn+1]. Then L(s) ≥ bγ√s log sc for all s ≥ t0.

Now, τX(s)v = τX(s)

(g − 1

|ΛK |∑

y∈ΛKJ

(yK

)τy (Gkg)

). Consider the support for this

function on the interval [tn, tn+1]. Since X(s) is constant on the interval, the individual

components suggest that support is contained within a cube that is centered at the origin

and having side length equal to |X(tn)|+ K + k + sg.

Choose t0 such that sg, the support of g is contained in Λb√t0/8c. We may need to

increase the value of t0, as outlined below, but this will be our starting point. Taking

into account the definition of X, K, k, then the support of τX(s)v is contained in a cube

having side length√

tn/4 + t(1−ε)/2n + t2ε

n +√

t0/8. Therefore,

s2τX(s)v

≤ 2tn/16 + 4t1−εn + 8t4ε

n + 8t0/64.


By definition of tn,

s2τX(s)v

≤ Rn0 t0/8 + 4(Rn

0 t0)1−ε + 8(Rn

0 t0)4ε + (Rn

0 t0)/8,

and assuming that ε < 1/5,

s2τX(s)v

≤ Rn0 t0/4 + 12(Rn

0 t0)1−ε.

As long as 12(Rn0 t0)

1−ε < 3Rn0 t0/4, then s2

τX(s)v≤ tn. If we set, t0 > (16)ε−1

, we have that

s2τX(s)v

≤ tn.

By setting γ ≥ (d + 2)/2, and utilizing the result of Proposition 4.1.1 we may then

show that the first term of (5.3.9) is bounded above by C(ρ, d) log

(t

t0

)< g2 >, for all

t ≥ t0, where we have bounded the variance of τX(tn)g− 1|ΛK |

∑y∈ΛK

J(

yK

)τy (Gkg) by

that the variance of g times some finite constant.

This leaves the second term in (5.3.9). We will introduce a further approximation

and employ the spectral gap result (Theorem 3.2.2), in an effort to control this term. In

particular, the second term in (5.3.9) depends upon the behaviour of the process in ΛL.

We wish to replace this with a function that depends only upon the total charge within

subsets of ΛL.

Step 2. Spectral Gap

Now apply spectral gap for the dynamics restricted to finite boxes in order to replace

GΛLτX(s)vs by a function that depends on the density of the charge in boxes of length

O(√

s).

Let l = l(s) = b√

2tn/(d + 2)R0c for s in the interval [tn, tn+1]. We shall assume

that t0 ≥ 2(d + 2)R0, in order to insure that l ≥ 2. Let R = (2l + 1)x, x ∈ Zd and

consider an enumeration of this set: R = x1, x2, . . . such that |xj| ≤ |xk| for j ≤ k.

Let Ωj = xj + Λl and let Mj = Mj(η) be the total charge in Ωj for the configuration η,

Mj =∑x∈Ωj

ηx.


Let q represent the total number of cubes with nonvoid intersection with ΛL and note

that q = O((L/l)d). For each j = 1, 2, . . . , denote by Mj the vector (M1, . . . , Mj).

Given a function h ∈ L2(νρ), Gl,Lh denotes the conditional expectation of h given Mq,

Gl,Lh = Eνρ [h|M1, . . . , Mq].

If l and L are chosen such that (2L+1)/(2l+1) is odd, then the tower property indicates

that

Gl,LGΛLh = Gl,Lh.

Given the flexibility in choosing L we may assume that (2L+1)/(2l+1) is odd. Therefore,

⟨(GΛL

h)2⟩

=⟨(Gl,Lh)2

⟩+

⟨(GΛL

h−Gl,Lh)2⟩.

Now we wish to show that the spectral gap result may be used to bound the second

term in the previous expression by the Dirichlet form, keeping in mind that we have

the integral involving −2D(νρ, vs) from our original construction (5.3.7) to absorb this

quantity. By Lemma 4.2.1 we have that

Eνρ

[(GΛL

h−Gl,Lh)2] ≤ R0l

2DΛL(νρ, h),

for any h ∈ L2(νρ). For any time tn ≤ s ≤ tn+1, l(s)2 ≤ 2s/(d + 2). So at this point,

t(d+2)/2⟨v2

t

⟩− t(d+2)/20

⟨v2

t0

⟩

= −2

∫ t

to


(d + 2

2

) ∫ t

to

sd/2⟨(τX(s)vs)

2⟩ds

= −2

∫ t

to


(d + 2

2

) ∫ t

to

sd/2⟨(τX(s)vs −GΛL

τX(s)vs)2⟩ds

+

(d + 2

2

) ∫ t

to

sd/2⟨(GΛL

τX(s)vs)2⟩ds

≤ −2

∫ t

to

s(d+2)/2D(νρ, τX(s)vs)ds + C (ρ, d) log

(t

t0

) ⟨g2

⟩ds

+

(d + 2

2

) ∫ t

to

sd/2⟨(GΛL

τX(s)vs −Gl,LτX(s)vs)2⟩ds

+

(d + 2

2

) ∫ t

to

sd/2⟨(Gl,LτX(s)vs)

2⟩ds


≤ −2

∫ t

to

s(d+2)/2D(νρ, τX(s)vs)ds + C (ρ, d) log

(t

t0

) ⟨g2

⟩ds

+

∫ t

to


(d + 2

2

) ∫ t

to


2⟩ds

≤(

d + 2

2

) ∫ t

to


2⟩ds + C (ρ, d) log

(t

t0

) ⟨g2

⟩(5.3.10)

for t0 ≥ 2(d + 2) and for all t ≥ t0.

Step 3. Space Averages

The previous formula holds for all trajectories X : R+ −→ Zd that are constant in

the interval [tn, tn+1] and such that |X(tn)| ≤ (1/4)√

tn. Recall that l(s) ≤ √tn by

construction. Therefore,

t(d+2)/2⟨v2

t

⟩− t(d+2)/20

⟨v2

t0

⟩

≤(

d + 2

2

) ∫ t

to

sd/2⟨(Gl,Lτxvs)

2⟩ds + C (ρ, d) log

(t

t0

) ⟨g2

⟩,

for all x ∈ Λl/4. Hence, we may average over x ∈ Λl/4 and show that

t(d+2)/2⟨v2

t

⟩− t(d+2)/20

⟨v2

t0

⟩

≤(

d + 2

2

) ∫ t

to

Avgx∈Λl/4

[sd/2

⟨(Gl,Lτxvs)

2⟩]

ds + C (ρ, d) log

(t

t0

) ⟨g2

⟩,

(5.3.11)

for all t ≥ t0.

Now, using the definition of vs and the Schwarz inequality,

(Gl,Lτxvs)2 =

(Gl,Lτx

(gs − Avg

y∈ΛK

[J

( y

K

)τy (Gkg)s

]))2

≤ 2

(Gl,Lτx

(gs − Avg

y∈ΛK

[J

( y

K

)τygs

]))2

+ 2

(Avgy∈ΛK

[J

( y

K

)Gl,Lτx+y (gs − (Gkg)s)

])2

.


Substituting into (5.3.11) we have that

t(d+2)/2⟨v2

t

⟩− t(d+2)/20

⟨v2

t0

⟩

≤∫ t

to

(d + 2) Avgx∈Λl/4

[sd/2

⟨(Gl,Lτx

(gs − Avg

y∈ΛK

[J

( y

K

)τygs

]))2⟩]

ds

+

∫ t

to

(d + 2) Avgx∈Λl/4

Avgy∈ΛK

J( y

K

)sd/2

⟨(Gl,Lτx+y (gs − (Gkg)s))

2⟩ ds

+C (ρ, d) log

(t

t0

) ⟨g2

⟩.

(5.3.12)

Now consider the first term in (5.3.12). Employing Schwarz inequality we may show that⟨(Gl,Lτx

(gs − Avg

y∈ΛK

[J

( y

K

)τygs

]))2⟩

≤ Avgy∈ΛK

[J

( y

K

)]Avgy∈ΛK

[J

( y

K

) ⟨(Gl,L (τxgs − τx+ygs))

2⟩] ,

where Avgy∈ΛK

[J

( y

K

)]−→ 1 as K −→∞ which is to say as s −→∞. Hence, proof of the

proposition relies upon two claims,∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L (τxgs − τx+ygs))

2⟩]]

ds ≤ Ct1−α (5.3.13)

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2


2⟩]]

ds ≤ Ct1−α, (5.3.14)


These two claims, which are addressed in subsequent lemmas, complete the proof of

the proposition.

Therefore, we require the following two lemmas.

Lemma 5.3.4.∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2


2⟩]]

ds ≤ Ct1−α,


Lemma 5.3.5.∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2


2⟩]]

ds ≤ Ct1−α,



The proof of each lemma proceeds in a similar fashion. We first wish to show that

the Ginzburg-Landau process on Zd is reasonably well approximated by a process that

includes the Ginzburg-Landau dynamics, plus a small Glauber component. We are in-

troducing the Glauber component to facilitate the entropy argument that is to follow in

Section 5.4. As discussed in Landim and Yau [LY03], having focussed attention on the

process acting on finite boxes with fixed charge on each box, we wish to consider the

associated product measure with respect to either the grand canonical measure, or the

canonical measure. In the end it is simpler to work with the grand canonical measure,

and the Glauber component is introduced in order to turn a singular measure with re-

spect to the grand canonical measure νρ into an absolutely continuous measure after any

positive time.

For δ > 0 denote

LG,δ =δ

2

∑

x∈Zd

∂2η(x) +

δ

2

∑

x∈Zd

[Φ(ρ)− V ′(η(x))] ∂η(x). (5.3.15)

Now, define hδs as P δ

s h, where P δs is the semigroup associated with L+LG,δ. The following

proposition and proof are due to Landim and Yau, [see [LY03]], with infinite volume

dynamics replacing the finite volume dynamics of the original.

Proposition 5.3.6. Assume that the potential V has a bounded second derivative,

‖V ′′‖ < ∞. Fix δ > 0 and a function h in L2(νρ) with finite triple norm. There exists a

constant C0, depending only on the potential V and the dimension d, such that

Eνρ [(Pth− P δt h)2] ≤ C0|||h|||2δt(1 + t),

for all t ≥ 0.

Proof. As before, let 〈 · 〉 represent expectation with respect to νρ. Let


S(t) = Sδ(t) = 〈(Pth− P δt h)2〉. S is a positive function and S(0) = 0.

S ′(t) = 2〈Pth− P δt h,LPth− (L+ LG,δ)P δ

t h〉

= 2〈Pth,LPth〉 − 2〈Pth,LP δt h〉 − 2〈Pth,LG,δP δ

t h〉

− 2〈P δt h,LPth〉+ 2〈P δ

t h,LP δt h〉+ 2〈P δ

t h,LG,δP δt h〉.

Since 〈(Pth− P δt h),L(Pth− P δ

t h)〉 ≤ 0, we have

2〈Pth,LPth〉 − 2〈Pth,LP δt h〉 − 2〈P δ

t h,LPth〉+ 2〈P δt h,LP δ

t h〉 ≤ 0.

Thus, S ′(t) ≤ 2〈P δt h,LG,δP δ

t h〉 − 2〈Pth,LG,δP δt h〉.

Similarly, since 〈(Pth− P δt h),LG,δ(Pth− P δ

t h)〉 ≤ 0, we have

−2〈Pth,LG,δP δt h〉 ≤ −〈Pth,LG,δPth〉 − 〈P δ

t h,LG,δP δt h〉,

meaning S ′(t) ≤ 〈P δt h,LG,δP δ

t h〉 − 〈Pth,LG,δPth〉. Finally, as the first term is less than,

or equal to 0, we have that

S ′(t) ≤ −〈Pth,LG,δPth〉.

By the definition of LG,δ (5.3.15),

−〈Pth,LG,δPth〉 =δ

2

∑

x∈Zd

Eνρ

[(∂Pth

∂η(x)

)2]

=δ

2

∑

x∈Zd

⟨(∂Pth

∂η(x)

)2⟩

Let R(t) =∑

x∈Zd

⟨(∂Pth

∂η(x)

)2⟩

. Then,

R′(t) = 2∑

x∈Zd

⟨∂Pth

∂η(x),∂LPth

∂η(x)

⟩= 2

∑

x∈Zd

⟨∂ht

∂η(x),

∂Lht

∂η(x)

⟩

If we once again consider nearest neighbour bonds b = (y, z), then

R′(t) = 2∑

x∈Zd

⟨∂ht

∂η(x),

∂Lht

∂η(x)

⟩= 2

∑

x∈Zd

⟨∂ht

∂η(x),∂

∑b Lbht

∂η(x)

⟩.


By the definition of Lb (4.1.1), if x 6= y and x 6= z then

⟨∂ht

∂η(x),∂Lbht

∂η(x)

⟩=

⟨∂ht

∂η(x),Lb

∂ht

∂η(x)

⟩≤ 0.

If x = y, or x = z, then

⟨∂ht

∂η(x),∂Lbht

∂η(x)

⟩=

⟨∂ht

∂η(x),Lb

∂ht

∂η(x)

⟩+

1

2Eνρ

[∂ht

∂η(x)V ′′(η(x))

(∂ht

∂η(y)− ∂ht

∂η(x)

)]

≤ 1

2Eνρ

[∂ht

∂η(x)V ′′(η(x))

(∂ht

∂η(y)− ∂ht

∂η(x)

)].

Therefore,

R′(t) ≤∑x∼y

Eνρ

[∂ht

∂η(x)V ′′(η(x))

(∂ht

∂η(y)− ∂ht

∂η(x)

)].

Since 2ab ≤ A−1a2 + Ab2, for any A > 0,

R′(t) ≤∑x∼y

Eνρ

[∂ht

∂η(x)V ′′(η(x))

(∂ht

∂η(y)− ∂ht

∂η(x)

)]

≤ A−1∑x∼y

Eνρ

[(∂ht

∂η(x)

)2]

+ A∑x∼y

Eνρ

[(V ′′(η(x))

(∂ht

∂η(y)− ∂ht

∂η(x)

))2]

≤ A−1R(t) + CA∑x∼y

Eνρ

[(∂ht

∂η(y)− ∂ht

∂η(x)

)2]

, (5.3.16)

by the definition of R(t) and since we have assumed that V ′′ is bounded, where C is some

finite constant which depends upon ‖V ′′‖∞. Since

∑x∼y

Eνρ

[(∂ht

∂η(y)− ∂ht

∂η(x)

)2]

= 〈ht, (−L)ht〉 ,

and ∂t 〈ht, ht〉 = 2 〈ht, (L)ht〉, we have

∫ t

0

〈hs, (−L)hs〉 ds = −1

2

∫ t

0

∂s 〈hs, hs〉 ds

=1

2〈h, h〉 − 1

2〈ht, ht〉

≤ 〈h, h〉 .


Recalling the previous bound on R′(t) (5.3.16) we have,

R(t)−R(0) =

∫ t

0

R′(s) ds

≤ A−1

∫ t

0

R(s) ds + CA

∫ t

0

∑x∼y

Eνρ

[(∂hs

∂η(y)− ∂hs

∂η(x)

)2]

ds

≤ A−1

∫ t

0

R(s) ds + CA

∫ t

0

〈hs, (−L)hs〉 ds

≤ A−1

∫ t

0

R(s) ds + CA 〈h, h〉 .

Minimizing over A,

R(t) ≤ R(0) + C ‖h‖2

(∫ t

0

R(s) ds

)1/2

,

for all t ≥ 0. This inequality gives us that

R(t) ≤ C (R(0) + 〈h, h〉 t) .

Now, consider 〈h, h〉. The following construction, appearing in [LY03], may be used to

show that 〈h, h〉 ≤ C|||h|||, for some finite constant C. Without loss of generality, we will

assume that Eνρ [h] = 0.

Let xj, j ≥ 1 be an enumeration of Zd and for j ≥ 1 denote by Fj the σ-algebra

generated by η(xi), 1 ≤ i ≤ j and let hj = Eνρ [h|Fj], with h0 = Eνρ [h]. Then

Eνρ [h2] =

∑j≥0

Eνρ [(hj − hj+1)2].

We may write Eνρ [(hj −hj+1)2] as Eνρ [Eνρ [(hj −hj+1)

2|Fj]], where the inner, conditional

expectation, Eνρ [(hj − hj+1)2|Fj], is equal to E

η(xj+1)νρ [(hj − hj+1)

2], where the latter

expectation indicates that variables η(x1), η(x2), . . . , η(xj) are considered constants

with respect to integration, with only η(xj+1) varying. Therefore, Eη(xj+1)νρ [(hj − hj+1)

2]

reflects the variance of hj+1 as a function of a single variable. The spectral gap for

Glauber dynamics allows us to bound the variance with the Dirichlet form, specifically

Eη(xj+1)νρ

[(hj − hj+1)2] ≤ CEη(xj+1)

νρ

[(∂hj+1

∂η(xj+1)

)2]

.


Since,

∂hj+1

∂η(xj+1)=

∂Eνρ [h |Fj+1 ]

∂η(xj+1)= Eνρ

[∂h

∂η(xj+1)

∣∣∣∣Fj+1

],

we have,

Eνρ [(hj − hj+1)2 | Fj] ≤ CEνρ

[(∂h

∂η(xj+1)

)2∣∣∣∣∣Fj+1

].

Therefore,

Eνρ [h2] ≤ C

∑j≥1

Eνρ

[(∂h

∂η(xj)

)2]≤ C|||h|||2,

by the definition of the triplenorm, for some finite constant C.

Gathering our results,

S ′(t) ≤ Cδ|||h|||2(1 + t),

and

S(t) = 〈(Pth− P δt h)2〉 ≤ Cδ|||h|||2t(1 + t)

as desired.

This result will reduce the proof of Lemma 5.3.4 and Lemma 5.3.5 to showing that

terms such as Eνρ

[(Gl,L

(P δ

s h))2

]decay sufficiently quickly. This will rely upon an

entropy argument which we pursue in the next section.

5.4 Entropy estimates

Recall

L =1

2

∑x∼y

(∂η(x) − ∂η(y))2 − 1

2

∑x∼y

(V ′(η(y))− V ′(η(x))) (∂η(y) − ∂η(x)),

and

LG,δ =δ

2

∑

x∈Zd

∂2η(x) +

δ

2

∑

x∈Zd

[Φ(ρ)− V ′(η(x))] ∂η(x),

for δ > 0.

Given the choice of the drift term, νρ is the unique invariant measure for the diffusion

process with generator L + LG,δ. Denote by P δt , t ≥ 0 the semigroup associated with


Lδ = L+LG,δ. Then, although νΛL,Mq is singular with respect to νρ, the addition of the

Glauber dynamics means that νΛL,MqPδt is absolutely continuous with respect to νρ for

any t > 0.

For t > 0 let ft = fδ,l,L,Mq

t be the Radon-Nikodym derivative of νΛL,MqPδt with respect

to νρ,

ft = fδ,l,L,Mq

t =dνΛL,MqP

δt

dνΛLρ

. (5.4.1)

Since νρ is translation invariant and reversible, and since the dynamics is translation

invariant

Gl,Lτxhs = Eνρ [τxhs|Mq] = Eνρ [hτ−xfs] . (5.4.2)

With the exception of the introduction of the Glauber dynamics, the entropy argument

in the case of the Ginzburg-Landau process follows that of the zero-range process. In

particular, the subsequent results appear as Lemma 4.3 and Lemma 4.4 in Janvresse,

Landim, Quastel and Yau [JLQY99].

Lemma 5.4.1. Let h be any local function. Let Λh be the smallest cube centred at the

origin containing the support of h− τeh for all unit vectors e in Zd. Then for f ≥ 0 with∥∥√f

∥∥2

= 1, there exists a finite constant C = C(ρ, h) depending only upon h and ρ such

that for any unit vector e in Zd

(Eνρ [f (h− τeh)]

)2 ≤ C(ρ, h)DΛh

(νρ,

√f)

where DΛhis the Dirichlet form associated with LΛh

Proof. Consider the Ginzburg-Landau process which corresponds to the Dirichlet form

DΛh

(νρ,√

f). The spectral gap of this process has magnitude Γ which is dependent

upon the size of the support of h. Let β ≤ (1/8)(‖h‖∞Γ)−1. Then by the perturbation

theorem, Theorem 3.2.3,

Eνρ [f (h− τeh)]− 1

βDΛh

(νρ,

√f)≤ 2β〈(h− τeh) , (−LΛh

)−1 (h− τeh)〉,


with LΛhbeing the generator restricted to the cube Λh with reflecting boundary condi-

tions. For β ≥ (1/8)(‖h‖∞Γ)−1, then Eνρ [f (h− τeh)] is bounded above by

2‖h‖∞ ≤ 16β‖h‖2∞. By optimizing over β the lemma is achieved.

Lemma 5.4.2. There exists finite constant C = C(d, h, ρ) depending only upon h,d and

ρ such that for n sufficiently large

1

|Λn|2∑

x,y∈Λh

(Eνρ [τxf (h− τyh)]

)2 ≤ C(d, h, ρ)n2−dDΛ3n

(νρ,

√f)

.

Proof. Define a canonical path from the origin to the point y, 0 = x0, x1, . . . , xm = y,

where m =∑d

j=1 |yj|. The path proceeds in a series of nearest neighbour steps, by

initially moving towards y in the first coordinate direction, followed by a move in the

second coordinate direction, and so on. Let ei = xi+1−xi be the unit vector representing

the i + 1st nearest neighbour step. Using this canonical path,

h− τyh =m−1∑i=0

τxih− τxi+1

h.

Thus, for any f ,

(Eνρ [f(h− τyh)]

)2=

(m−1∑i=0

Eνρ [τ−xif(h− τeh)]

)2

.

By Schwarz’s inequality and the previous lemma,

(m−1∑i=0

Eνρ [τ−xif(h− τeh)]

)2

≤ C(d, h, ρ)mm−1∑i=0

DΛh

(νρ,

√τ−xi

f)

.

For n larger than the side length of Λh we have that m ≤ 2dn. Also, we have

DΛh

(νρ,√

τxf)

= Dτ−xΛh

(νρ,√

f). By counting explicity

1

|Λn|2∑

x,y∈Λh

(Eνρ [τxf(h− τyh)]

)2 ≤ C(d, h, ρ)n2−dDΛ3n

(νρ,

√f)

involving some new constant C(d, h, ρ).


The proof of Lemma 5.3.4 will rely upon an entropy result to follow. Two prelimi-

nary results are required which may be found in [LY03] and are repeated below. Whereas

Landim and Yau consider the generator restricted to ΛL, we are working with the gen-

erator in infinite volume in what follows. However, this change has no impact upon the

necessary results and only negligible impact upon the proofs of those results. We have

chosen to include the additional detail in the proof of Lemma 5.4.3 in order to show that

the move from finite volume to infinite volume is entirely absorbed by the constant in

the desired inequality. The same construction could be employed in the proof of Lemma

5.4.4 with the result that once again the move from finite volume to infinite volume is

entirely absorbed by the constant in the desired inequality. However, we have chosen to

omit this additional detail in order to emphasize the key elements of the Landim and

Yau [LY03] result.

Lemma 5.4.3. For t ≥ 0, let

W (t) = EΛL,Mq

[ ∑x∈ΛL

ηt(x)2

].

Then W (t) ≤ W (0)eC1t + Ld(eC1t − 1), for all t ≥ 0 for some finite constant C1, that

depends upon d and ‖F ′‖∞ from assumption A1.

Proof. To begin, note that,

Lδ∑x∈ΛL

η(x)2 =(L+ LG,δ

) ∑x∈ΛL

η(x)2

As before, the generator L may be written as the sum of the generator over nearest

neighbour bonds b = (x, y).

(Lh) =∑

b

(Lbh),

where

(Lbh) = (∂η(x) − ∂η(y))2h(η)− (V ′(η(y))− V ′(η(x)))(∂η(y) − ∂η(x))h(η), (5.4.3)


A bond b = (x, y) is said to belong to ΛL if both x and y belong to ΛL. A bond

b = (x, y) is said to belong to ∂ΛL, the boundary of ΛL, if exactly one of x or y belongs

to ΛL. In such cases x will be identified as the end that belongs to ΛL. Therefore,

(L+ LG,δ) ∑

x∈ΛL

η(x)2 =1

2

∑

b∈ΛL

(∂η(y) − ∂η(z))2

∑x∈ΛL

η(x)2

− 1

2

∑

b∈ΛL

(V ′(η(z))− V ′(η(y))) (∂η(z) − ∂η(y))∑x∈ΛL

η(x)2

+1

2

∑

b∈∂ΛL

(∂η(y) − ∂η(z))2

∑x∈ΛL

η(x)2

− 1

2

∑

b∈∂ΛL

(V ′(η(z))− V ′(η(y))) (∂η(z) − ∂η(y))∑x∈ΛL

η(x)2

+δ

2

∑

y∈Zd

∂2η(y)

∑x∈ΛL

η(x)2 +δ

2

∑

y∈Zd

[Φ(ρ)− V ′(η(y))] ∂η(y)

∑x∈ΛL

η(x)2

≤ 8d |ΛL| −∑

b∈ΛL

(V ′(η(y))− V ′(η(x))) (η(y)− η(x))

∑

b∈∂ΛL

(V ′(η(y))− V ′(η(x))) η(x)

+ δ |ΛL|+ δΦ(ρ)∑x∈ΛL

η(x)− δ∑x∈ΛL

V ′(η(x))η(x)

Since 2ab ≤ a2 + b2,δ ≤ 1 and, by assumption (A1), V ′(a)2 ≤ C0(1 + a2), where C0

depends upon ‖F ′‖∞.

(i) δΦ(ρ)∑x∈ΛL

η(x) ≤ Φ(ρ)

(|ΛL|+

∑x∈ΛL

η(x)2

)

(ii) δ∑x∈ΛL

V ′(η(x))η(x) ≤ δ∑x∈ΛL

(V ′(η(x))2 + η(x)2

)

≤ C0 |ΛL|+ (C0 + 1)∑x∈ΛL

η(x)2


(iii)∑

b∈ΛL

(V ′(η(y))− V ′(η(x))) (η(y)− η(x))

≤∑

b∈ΛL

(V ′(η(y))2 + V ′(η(x))2 + η(y)2 + η(x)2

)

≤ 4d∑x∈ΛL

(V ′(η(x))2 + η(x)2

)

≤ 4dC0 |ΛL|+ 4d(C0 + 1)∑x∈ΛL

η(x)2

(iv)∑

b∈∂ΛL

(V ′(η(y))− V ′(η(x))) η(x)

≤∑

b∈∂ΛL

(V ′(η(y))2 + V ′(η(x))2 + η(x)2

)

=∑

b∈∂ΛL

V ′(η(y))2 +∑

b∈∂ΛL

(V ′(η(x))2 + η(x)2

)

≤∑

b∈∂ΛL

η(y)2 + 8dC0 |ΛL|+ 4d(C0 + 1)∑x∈ΛL

η(x)2

Note, in (iv) we can bound EΛL,Mq

[∑b∈∂ΛL

η(y)2]

by C1 |ΛL| where C1 depends upon ρ.

Therefore,

Lδ∑x∈ΛL

η(x)2 ≤ 8d |ΛL|+ 4dC0 |ΛL|+ 4d(C0 + 1)∑x∈ΛL

η(x)2

+∑

b∈∂ΛL

η(y)2 + 8dC0 |ΛL|+ 4d(C0 + 1)∑x∈ΛL

η(x)2

+ |ΛL|+ Φ(ρ)

(|ΛL|+

∑x∈ΛL

η(x)2

)+ C0 |ΛL|+ (C0 + 1)

∑x∈ΛL

η(x)2

≤ C1

(Ld +

∑

b∈∂ΛL

η(y)2 +∑x∈ΛL

η(x)2

),

where C1 depends upon ρ, d and C0. Taking expectations we see that W ′(t) ≤ C1

(Ld + W (t)

)

for some C1 that depends upon ρ, d and C0 and by applying Gronwall’s inequality

W (t) ≤ W (0)eC1t + Ld(eC1t − 1).

Notice from the statement of the previous lemma that, for t ≤ 1

W (t) ≤ W (0)eC1t + Ld(eC1t − 1

) ≤ C2

(W (0) + tLd

)(5.4.4)


Lemma 5.4.4. For t ≥ 0 and Mk =∑

x∈Ωkη(x) let

U(t) =

q∑

k=1

EΛL,Mq

[(Mk(t)−Mk)

2] .

Then U(t) ≤ ld−1(W (0) + Ld

)(eC1t − 1), for all 0 ≤ t ≤ 1 for some finite constant C1,

that depends upon d and ‖F ′‖∞ from assumption A1.

Proof. To begin, note that for any smooth function H,

∂η(x)H(Mk) = 1x ∈ Ωk∂MkH(Mk).

We can therefore show that

Lδ

q∑

k=1

M2k =

(L+ LG,δ) q∑

k=1

M2k

=1

2

∑x∼y

(∂η(x) − ∂η(y))2

q∑

k=1

M2k

− 1

2

∑x∼y

(V ′(η(y))− V ′(η(x))) (∂η(y) − ∂η(x))

q∑

k=1

M2k

+δ

2

∑

x∈Zd

∂2η(x)

q∑

k=1

M2k +

δ

2

∑

x∈Zd

[Φ(ρ)− V ′(η(x))] ∂η(x)

q∑

k=1

M2k

=|ΛL||Λl|

(δ + 4d |Λl|(d−1)/d

)−

∑

k∼j

(Mk −Mj)(V ′

k,j − V ′j,k

)

− δ

q∑

k=1

Mk

∑x∈Ωk

V ′(η(x)).

Here V ′k,j stands for

∑x V ′(η(x)), where the summation takes place over all sites x in Ωk

that are distance 1 from Ωj. The factor ld−1 in the first term comes from the fact that

∑qk=1 M2

k changes only due to diffusion at the boundary of the squares Ωi, 1 ≤ i ≤ q.Since δ ≤ l−1, 2ab ≤ a2 + b2, and by assumption (A1), V ′(a)2 ≤ C0(1 + a2), where C0

depends upon ‖F ′‖∞, we can bound the previous expression by

C1

[Ldld−1 +

q∑

k=1

M2k + ld−1

∑x∈ΛL

η(x)2

].


From this we see that U ′(t) ≤ C1

[Ldld−1 + U(t) + ld−1W (t)

]. By (5.4.4), W (t) ≤

C2

(W (0) + tLd

), for t ≤ 1. Hence, U ′(t) ≤ C1

[Ldld−1 + ld−1W (0) + U(t)

]. By ap-

plying Gronwall’s inequality U(t) ≤ U(0)eC1t + (Ldld−1 + ld−1W (0))(eC1t − 1). Since

U(0) = 0, U(t) ≤ ld−1(Ld + W (0))(eC1t − 1).

As before note that, given the statement of the previous lemma, there exists a finite

constant C2, such that for t ≤ 1,

U(t) ≤ C2ld−1

(W (0) + Ld

)t. (5.4.5)

We now move to the desired entropy result. The result, albeit with a slightly different

definition of the density fδ,l,L,Mq

t , appears in [LY03]. In [LY03], the definition of definition

of fδ,l,L,Mq

t makes use of the finite volume dynamics, P δ,ΛLt , whereas we continue to employ

infinite volume dynamics, P δt , in what follows. However, the change from the original

result, replacing finite volume dynamics with infinite volume dynamics, has little impact

upon the proof which carries through in much the same manner, and is reproduced below

for completeness. Denote by H(f) the relative entropy of fdνρ with respect to νρ where

f is a density with respect to νρ.

Proposition 5.4.5. With fδ,l,L,Mq

t defined as before (5.4.1), and assuming the potential

V satisfies (A1), then there exists a finite constant C0 = C0(ρ, d, V ) such that

Eνρ

[H

(f

δ,l,L,Mq

t

)]≤ C0

(L

l

)d (log l + log δ−1

)

for all t ≥ 1. Expectation is with respect to Mq.

Proof. Fix a vector Mq = (M1, . . . , Mq) and recall

fδ,l,L,Mq

t =dνΛL,MqP

δt

dνΛLρ

,

where νΛL,Mq represents the measure dνΩ1,M1 ⊗ · · · ⊗ dνΩq ,Mq , and P δt is the semigroup

associated with L+ LG,δ as defined in (3.1.1) and (5.3.15). Define the variables

Nk =∑

x∈Ωkη(x), for k = 1, . . . q, and define Nq = (N1, . . . , Nq).


Since νΛL,Mq is not absolutely continuous with respect to νρ, the approach of the proof

is to introduce a function gt = gt(Nq), that depends upon the total charge within each

cube Ωk, 1 ≤ k ≤ q. Since,

H(ft) =

∫ft log ft dνρ =

∫ft log

ft

gt

dνρ +

∫ft log gt dνρ, (5.4.6)

a careful choice of gt will allow us to achieve the desired bound. To begin, we will choose

gt, so that at time 0, gt is a Dirac measure at Mq. Since gt equals ft at time zero, the

first term will disappear at t = 0. The time derivative of the first term is bounded above

by (see [Yau91])∫

ftLδgt − ∂tgt

gt

dνρ.

In addition, our choice of gt should permit many of the terms in Lδgt − ∂tgt to cancel,

to the extent that the integral is small for small t. Given our choice of gt we also

require that the second integral in our decomposition is sufficiently small for small t.

Let 2K + 1 = (2L + 1)/(2l + 1). Denote by TdK , the set ΛK viewed as a d-dimension

torus with (2K + 1)d points and denote the points of TdK by the letters x = (x1, . . . , xd).

Let ∆ represent the discrete Laplacian on TdK , namely for any function f : Td

K → R,

(∆f)(x) =∑

1≤i≤d f(x + ei) + f(x − ei) − 2f(x), where ei, 1 ≤ i ≤ d is the canonical

basis for R. We may represent ∆ as a matrix with entries ∆x,y, x, y ∈ TdK, given by

∆x,x = −2d, ∆x,y = 1, for x ∼ y, and ∆x,y = 0 otherwise. Denote by X = Xδ,l =

(2l + 1)d−1(2l + 1)δI −∆, where I stands for the identity matrix. X, as defined, is a

strictly positive operator. Let Σ = X−1.

For t ≥ 0, define gt(Nq) as follows,

gt(Nq) =1

(|2πtΣ−1|)1/2exp

(− 1

2t[Nq −Mq]Σ[Nq −Mq]

),

where |Σ| denotes the determinant of Σ.

Using this definition of gt, we may bound the second term in the decomposition of


entropy (5.4.6),

∫ft log gt dνρ ≤ −1

2log

∣∣2πtΣ−1∣∣− 1

2tEΛL,Mq [(Mq(t)−Mq) Σ (Mq(t)−Mq)]

≤ −1

2log

∣∣2πtΣ−1∣∣,

since Σ is a positive operator. Also, given the definition of Σ,

∣∣2πtΣ−1∣∣ =

(2πt(2l + 1)d−1

)q |γI −∆| ,

where γ = δ(2l + 1). Since −∆ is a positive operator, the eigenvalues of γI − ∆ are

bounded below by γ, meaning |γI −∆| ≥ γq. Therefore, |2πtΣ−1| ≥ (2πt |Λl| δ)q. Hence,

the second term in the decomposition may be bounded above,

∫ft log gt dνρ ≤ − |ΛL|

2 |Λl| log 2πt |Λl| δ (5.4.7)

As noted previously, the time derivative of∫

ft log ft

gtdνρ is bounded above by

∫ftLδgt − ∂tgt

gt

dνρ.

To compute Lδgt, note that ∂η(x)gt = ∂Mjgt for x in Ωj. Given the definition of gt, the

second order operators of Lδgt cancel with the time derivative of gt, and we are left with

the drift terms,

(Lδ − ∂t

)gt(Nq)

gt(Nq)=

1

2t

∑

k∼j

(∑[N−M]

)k−

(∑[N−M]

)j

V ′

k,j − V ′j,k

(5.4.8)

+∑

k

(∑[N−M]

)k

∑x∈Ωk

V ′(η(x)).

Here, summation k ∼ j involves all indices k, j such that the cubes Ωk, Ωj are adjacent.

Also, recall that V ′k,j stands for

∑x V ′(η(x)), where the summation takes place over

all sites x in Ωk that are distance 1 from Ωj. Also, in order to simplify notation, the

chemical potential λ is assumed to be 0. This results in no loss of generality, since the


only assumption used in what follows is (A1), which is satisfied for −λa + V (a) as long

as it is satisfied for V (a). We now wish to establish that

EΛL,Mq

[(Lδ − ∂t

)gt(Mq(t))

gt(Mq(t))

]≤ C1

(δlt)1/2

(W (0) + Ld

),

for 0 ≤ t ≤ 1, C1 is some finite constant and W (t) = EΛL,Mq

[∑x∈ΛL

ηt(x)2].

Now considering the first term of (5.4.8), and making use of the fact that

2ab ≤ Aa2 + A−1b2, for A > 0

EΛL,Mq

[1

2t

∑

k∼j

(∑[M(t)−M]

)k−

(∑[M(t)−M]

)j

V ′

k,j − V ′j,k

]

≤ 1

4AtEΛL,Mq

[∑

k∼j

(∑[M(t)−M]

)k−

(∑[M(t)−M]

)j

2]

(5.4.9)

+2dA(2l + 1)d−1

t

∑x∈ΛL

EΛL,Mq

[V ′(ηt(x))2

]

for every A > 0. A Fourier computation shows that the first term in (5.4.9) is bounded

above by

C1l1−2d

Aδt

q∑

k=1

EΛL,Mq

[(Mk(t)−Mk)

2] ,

for some finite constant C1 that depends on d alone. By (5.4.5), the sum in this term is

bounded above by C2ld−1

(W (0) + Ld

)t, meaning that the entire term is bounded above

by C2

(W (0) + Ld

)/Aδld. Similarly, using assumption (A1) and (5.4.5), the second term

in (5.4.9) is bounded above by C2A(W (0) + Ld

)/δld−1t. Minimizing over A, we have

that (5.4.9) is bounded above by C2

(W (0) + Ld

)/δlt.

Turning to the second term of (5.4.8), we have, making use of the fact that

2ab ≤ Aa2 + A−1b2, that

EΛL,Mq

[∑

k

(∑[M(t)−M]

)k

∑x∈Ωk

V ′(η(x))

]≤δA

4t

∑

k

EΛL,Mq

[(∑[M(t)−M]

)k

2]

(5.4.10)

+C2δl

d

4At

∑x∈ΛL

EΛL,Mq

[V ′(ηt(x))2

]


A Fourier computation and assumption (A1) allows us to bound this above by

C2A

δl2dt

q∑

k=1

EΛL,Mq

[(Mk(t)−Mk)

2] +C2δl

d

At

(Ld

∑x∈ΛL

EΛL,Mq

[ηt(x)2

])

.

By (5.4.4) and (5.4.5), this is bounded above by

C2A

δld+1

(Ld + W (0)

)+

C2δld

At

(Ld + W (0)

)

Minimizing over A we have that (5.4.10) is bounded above by

C2

lt

(Ld + W (0)

)

Therefore, in light of (5.4.8), (5.4.9), (5.4.10) and the associated calculations, we have

EΛL,Mq

[(Lδ − ∂t

)gt(Mq(t))

gt(Mq(t))

]≤ C1

(δlt)1/2

(W (0) + Ld

),

for 0 ≤ t ≤ 1, C1 is some finite constant and W (t) = EΛL,Mq

[∑x∈ΛL

ηt(x)2]. Noting

that EΛL,Mq

[(Lδ−∂t)gt(Mq(t))

gt(Mq(t))

]is an upper bound for the time derivative of

∫ft log ft

gtdνρ,

then by integrating over time

EΛL,Mq

[(Lδ − ∂t

)gt(Mq(t))

gt(Mq(t))

]≤ C2

√t√

δl

(W (0) + Ld

).

Taking this result together with (5.4.6) and (5.4.7) we have that

Eνρ [H(ft)] ≤ C2

√t√

δl

(Eνρ [W (0)] + Ld

)− |ΛL|2 |Λl| log 2πt |Λl| δ

By the definition of W (t),

Eνρ [W (0)] = |Λl|q∑

k=1

Eνρ

[EΩk,Mk

[η(0)2

]]= |ΛL|Eνρ

[η(0)2

].

Hence,


√tLd

√δl

− |ΛL|2 |Λl| log 2πt |Λl| δ

Letting t∗ = δl1−2d,

Eνρ [H(ft∗)] ≤ C2

√t∗Ld

√δl

− |ΛL|2 |Λl| log 2πt∗ |Λl| δ

=C2

√δl1−2dLd

√δl

− |ΛL|2 |Λl| log

2πδl1−2d |Λl| δ

≤ C2

(L

l


)


Since l ≥ 1 and δ ≤ 1, we have that t∗ ≤ 1, and since H(ft) decreases in time,


(L

l


),

for all t ≥ 1, as desired.

5.5 Proof of Lemma 5.3.4 and Lemma 5.3.5

The proof of Lemma 5.3.4 and Lemma 5.3.5 will proceed similarly and will rely upon

some common elements.

To begin, we are introducing a Glauber component, defined in terms of some δ > 0.

In what follows, we will set δ = t−d/2−2−α, for some α > 0. We will also make use of

the Dirichlet form associated with Lδ = L+ LG,δ, the generator combining the Glauber

dynamics and the Ginzburg-Landau dynamics. That is, we will define

Dδ (ν, f) = −∫

fLδf dν

= −∫

f(L+ LG,δ

)f dν

= −∫

fLf dν −∫

fLG,δf dν.

We will make use of the fact that the Dirichlet form so defined provides an upper bound

on the Dirichlet form associated with the Ginzburg-Landau dynamics alone,

D (ν, f) ≤ Dδ (ν, f) ,

where

D (ν, f) = −∫

fLf dν.

Finally, since νρ is the invariant measure for the process having generator Lδ, we will

make use of the fact that∫ t

sDδ

(ν,√

fr

)dr is bounded above by H(fs), (see [KL99]

Theorem 9.2) with fs as defined in (5.4.1).


Lemma (5.3.4).

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2


2⟩]]

ds ≤ Ct1−α,

for some finite constant C, independent of γ, and α > 0.

Proof. Let h = τxg − τx+yg, so that Psh = Ps(τxg − τx+yg) = τxgs − τx+ygs. Then,

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2〈(Gl,L (Psh))2〉

]]ds

≤ 2

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(Psh− P δ

s h))2

⟩]]ds (5.5.1)

+ 2

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(P δ

s h))2

⟩]]ds. (5.5.2)

We will show that both (5.5.1) and (5.5.2) are bounded above by Ct1−α.

Consider (5.5.1), which is bounded above by

∫ t

t0

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Psh− P δ

s h)2

⟩]]ds.

By Proposition 5.3.6,

⟨(Psh− P δ

s h)2

⟩≤ C0|||h|||2δs(1 + s),

for δ > 0 and all s ≥ 0.

Thus the previous term is bounded above by

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2C0|||h|||2δs(1 + s)

]]ds.

and, since δ = t−d/2−2−α,

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Psh− P δ

s h)2

⟩]]ds ≤ Ct1−α.

Now consider (5.5.2). Fix an interval [tn, tn+1]. By definition, l(s) = b√

2tn/(d + 2)R0cand L(s) = bγ√tn+1 log tn+1c are constant in this interval [tn, tn+1]. Recalling (5.4.2),


since νρ is translation invariant and reversible, and since the dynamics is translation

invariant,

Gl,Lτx(Pδs g − τyP

δs g) = Eνρ [(g − τyg) τ−xfs] .

Using the method of Lemma 5.4.2,

Avgx∈Λl/4

[1

|ΛK |J( y

K

)Eνρ

[(Gl,Lτx(P

δs g − τyP

δs g)

)2]]

≤ C(d, g, ρ)K2

ldEνρ

[Dδ,Λl

(νρ,

√fs

)].

So,

∫ tn+1

tn

Avgx∈Λl/4

[1

|ΛK |J( y

K

)sd/2Eνρ

[(Gl,Lτx(P

δs g − τyP

δs g)

)2]]

ds

≤∫ tn+1

tn

sd/2C(d, g, ρ)K(s)2

l(s)dEνρ

[Dδ,Λl

(νρ,

√fs

)]ds,

and since l−d cancels sd/2 and K2(s) ≤ C(R0, d)t1−ε for s ≤ t this is bounded above by

t1−εC(d, g, ρ)Eνρ

[∫ tn+1

tn

Dδ,Λl

(νρ,

√fs

)]ds.

Since L > l, this is less than

t1−εC(d, g, ρ)Eνρ

[∫ tn+1

tn

Dδ,ΛL

(νρ,

√fs

)]ds.

As mentioned previously, given a density f with respect to νρ,

∫ t

s

Dδ,ΛL

(ν,

√fr

)dr ≤ H(fs).

Hence, the previous term is bounded above by

t1−εC(d, g, ρ)Eνρ [H (ftn)] .

Now, by Proposition 5.4.5

Eνρ

[H

(f

δ,l,L,Mq

tn

)]≤ C0

(L

l


).


Therefore, the previous term is bounded above by

t1−εC(d, g, ρ)

(L

l


).

Using the definitions of L , l and δ, this is, in turn, bounded above by

t1−εC(d, g, ρ)(log t)d+1.

Thus,

∫ tn+1

tn

Avgx∈Λl/4

[1

|ΛK |J( y

K

)sd/2Eνρ

[(Gl,Lτx(P

δs g − τyP

δs g)

)2]]

ds

≤ t1−εC(d, g, ρ)(log t)d+1.

It is left to sum over n, but recalling that t = tn(t)+1, where tn = Rn0 t0, we have that

n(t) ≤ log t. Therefore,

∫ tn(t)+1

t0

Avgx∈Λl/4

[1

|ΛK |J( y

K

)sd/2Eνρ

[(Gl,Lτx(P

δs g − τyP

δs g)

)2]]

ds

=

n(t)∑n=0

∫ tn+1

tn

Avgx∈Λl/4

[1

|ΛK |J( y

K

)sd/2Eνρ

[(Gl,Lτx(P

δs g − τyP

δs g)

)2]]

ds

≤n(t)∑n=0

[t1−εC(d, g, ρ)(log t)d+1

]

≤ [t1−εC(d, g, ρ)(log t)d+2

].

Hence, for ε > α > 0, and t sufficiently large, this is bounded above by C(d, g, ρ)t1−α, as

required.

Lemma (5.3.5).

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2


2⟩]]

ds ≤ Ct1−α,

for some finite constant C, independent of γ, and α > 0.


Proof. Let h = τx+y (g −Gkg), so that

Psh = Psτx+y (g −Gkg) = τx+y (gs − (Gkg)s) .

Then,∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2〈(Gl,L (Psh))2〉

]]ds

≤ 2

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(Psh− P δ

s h))2

⟩]]ds (5.5.3)

+ 2

∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(P δ

s h))2

⟩]]ds. (5.5.4)

As before, we will show that both (5.5.3) and (5.5.4) are bounded above by Ct1−α.

Considering (5.5.3),∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(Psh− P δ

s h))2

⟩]]ds

≤∫ t

t0

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Psh− P δ

s h)2

⟩]]ds.

By Proposition 5.3.6,

⟨(Psh− P δ

s h)2

⟩≤ C0|||h|||2δs(1 + s),

for δ > 0 and all s ≥ 0. Thus the previous term is bounded above by∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2C0|||h|||2δs(1 + s)

]]ds.

As δ = t−d/2−2−α,∫ t

to

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Psh− P δ

s h)2

⟩]]ds ≤ Ct1−α.

Now consider (5.5.4). Begin by fixing an interval [tn, tn+1].

Gl,LτxPδs h = Eνρ

[τxP

δs h|Mq

]=

∫τxP

δs h(η)f(η)νρ(dη) = Eνρ [τxhfs] .

Taken together with the fact that (g −Gkg) is FΛk-measurable,

Gl,Lτx+y

[P δ

s g − P δs (Gkg)

]= Eνρ

[(g −Gkg)τ−(x+y)fs

]

= Eνρ

[(g −Gkg)Gkτ−(x+y)fs

]


By the perturbation theorem, Theorem 3.2.3, and the spectral gap for Ginzburg

Landau dynamics, the latter expression may be bounded above by

β−1Eνρ

[DΛk

(νΛk

ρ ,√

τ−(x+y)fs

)]+ 2βEνρ

[(g −Gkg)(−LΛk

)−1(g −Gkg)]

for all β ≤ C/‖g‖∞k2. This relies upon the convexity of the Dirichlet form in order

to bound the expression DΛk

(νΛk

ρ , Gkf)

by DΛk

(νΛk

ρ , f).

For β ≥ C/‖g‖∞k2, since Eνρ


]is bounded above by 2‖g‖∞, we

have that

Eνρ


] ≤ β−1Eνρ

[DΛk

(νΛk

ρ ,√

τ−(x+y)fs

)]+ βk2C(g).

It is left to minimize over β, which produces

(Gl,Lτx+y

[P δ

s g − P δs h(Gkg)

])2 ≤ C(g, ρ)k2DΛk

(νΛk

ρ ,√

τ−(x+y)fs

),

which in turn is bounded above by

C(g, ρ)k2Dδ,Λk

(νΛk

ρ ,√

τ−(x+y)fs

),

Thus the time integral that appears in (5.5.4) when restricted to the time interval

[tn, tn+1] is bounded above by

C(g, ρ, d)A(J)Eνρ

[∫ tn+1

tn

sd/2k2

ldDδ,Λl

(νρ,

√fs

)ds

].

where A(J) = Avgy J(y/K). Here l−d cancels sd/2 and k(s)2 is bounded by t4ε, so

this is bounded above by

t4εA(J)C(d, g, ρ)Eνρ

[∫ tn+1

tn

Dδ,Λl

(νρ,

√fs

)]ds.

As before, since L > l, this is less than

t4εA(J)C(d, g, ρ)Eνρ

[∫ tn+1

tn

Dδ,ΛL

(νρ,

√fs

)]ds.


We now make use of the fact that the relative entropy provides an upper bound on the

Dirichlet form, ∫ t

s

Dδ,ΛL

(ν,

√fr

)dr ≤ H(fs)

to provide the following upper bound on the previous term

t4εA(J)C(d, g, ρ)Eνρ [H (ftn)] .

By Proposition 5.4.5

Eνρ

[H

(f

δ,l,L,Mq

tn

)]≤ C0

(L

l


).

Therefore, the previous term is bounded above by

t4εA(J)C(d, g, ρ)

(L

l


)

Using the definitions of L , l and δ, this is, in turn, bounded above by

t4εA(J)C(d, g, ρ)(log t)d+1.

Thus,

∫ tn+1

tn

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(P δ

s h))2

⟩]]ds ≤ t4εA(J)C(d, g, ρ)(log t)d+1.

It is left to sum over n, but recalling that t = tn(t)+1, where tn = Rn0 t0, we have that

n(t) ≤ log t. Therefore,

∫ tn(t)+1

t0

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(P δ

s h))2

⟩]]ds

=

n(t)∑n=0

∫ tn+1

tn

Avgx∈Λl/4

[Avgy∈ΛK

[J

( y

K

)sd/2

⟨(Gl,L

(P δ

s h))2

⟩]]ds

≤n(t)∑n=0

[t4εA(J)C(d, g, ρ)(log t)d+1

]

≤ [t4εA(J)C(d, g, ρ)(log t)d+2

].

Since A(J) ≈ 1, then for 1/5 > ε > α > 0, and sufficiently large t, this is bounded above

by C(d, g, ρ)t1−α, as required.


5.6 Rate of decay to equilibrium for general func-

tions

Theorem 5.6.1. Let g be a bounded, mean zero, local function, with finite triple norm.

Then for t > 0

Eνρ [(Ptg(η))2] =χ(ρ)(g′(ρ))2

[8πΦ′(ρ)t]d/2+ o(t−d/2), (5.6.1)


Proof. Given g, a mean-zero bounded local function, with finite triple norm, write

Eνρ

[(Pt(g(η)))2]

≤ (1 + A)Eνρ

(Pt

(g(η)− g′(ρ)

A(J)

1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

))2

+ (1 + A−1)Eνρ

(Pt

(g′(ρ)

A(J)

1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

))2 ,

where

A(J) =1

|ΛK |∑

x∈Zd

J( x

K

).

Let

U(η) = g(η)− g′(ρ)

A(J)

1

|ΛK |∑

x∈Zd

J( x

K

)τx Avg

y∈Λk

(η(y)− ρ)

Then U(ρ) = U ′(ρ) = 0. We have previously shown (see Theorem 5.3.3)that

Eνρ

[(Pt (U(η)))2] ≤ Ct−d/2−α,

for some finite constant C and α > 0. Since A(J) → 1 as t → ∞, then by Theorem

5.2.5, the second term equals

χ(ρ)(g′(ρ))2

[8πΦ′(ρ)t]d/2+ o(t−d/2).

Hence,

Eνρ

[(Pt(g(η)))2] =

χ(ρ)(g′(ρ))2

[8πΦ′(ρ)t]d/2+ o(t−d/2),

as desired.

Chapter 6

Central Limit Theorem:Ginzburg-Landau

6.1 Outline of Proof

Let us recall the decay estimate of Theorem 5.6.1

Eνρ [(Ptg(η))2] =[g′(ρ)]2χ(ρ)

[8πΦ′(ρ)t]d/2+ o(t−d/2).

This suggests that the general result of Kipnis and Varadhan may be used to establish

the central limit theorem and invariance principle for d ≥ 3. But what of d = 1 and

d = 2?

Note that since g′(ρ) = ∂αEνα [g]|α=ρ, the particular function, g under consideration

has little bearing upon the actual decay rate, entering only in the form of a constant

related to the mean value of the function. It seems reasonable, therefore, to approach the

problem by proving the invariance result first for some easy function and then extending

the result to more general functions. This was precisely the strategy employed by Quastel,

Jankowski and Sheriff [QJS02].

That leaves the choice of the easy function for which to establish the desired invariance

result. Here we take the martingale approach of [Kip87] and [QJS02]. The approach

taken by Kipnis [Kip87] was to identify some function U such that LU = η(0) − ρ.

Hence,∫ t

0(ηs(0) − ρ)ds is essentially a martingale and we may use a martingale central

78

Chapter 6. Central Limit Theorem: Ginzburg-Landau 79

limit theorem to prove that the rescaled quantity converges as desired. Unfortunately,

such a U is not readily apparent for models other than that of symmetric simple exclusion.

However, we will follow the spirit of this approach and represent∫ t

0V ′(ηs(0)) − Φ(ρ)ds

as a martingale plus an additive functional.

Specifically, we require a function U such that (∂s + L)U = V ′(ηs(0)) − Φ(ρ) plus

additional terms that, in the limit, are either well behaved, or trivial. This will allow us

to prove the following invariance result for our easy function, which is the subject of the

next section.

Lemma 6.1.1. Let

XNt =

1

a(N, d)

∫ Nt

0

V ′(ηs(0))− Φ(ρ)ds,

a(N, d) =

N3/4 d = 1√

N log N d = 2

N1/2 d ≥ 3

Let PN be the distribution of XNt under Pρ. Then PN are tight and has unique weak

limit

d = 1 : fractional Brownian motion,

cov.2Φ(ρ)

√Φ′(ρ)

3√

π[t

3/22 + t

3/21 − |t2 − t1|3/2],

d = 2 : Brownian motion, covarianceΦ(ρ)

2πmin(t1, t2).

The dynamics of the process will prove sufficient to ensure that the limiting process

is Gaussian, while the decay result establishes the desired covariance structure.

Having proved the result in the simple case, the final section of this chapter will

demonstrate that any mean zero local function may be approximated by a suitably

rescaled version of our easy function, in the sense that the difference converges to the

trivial process. This will therefore establish Theorem 3.3.2.


For both the simple case and general case, we will rely upon the following results.

The proof of the following lemma appears in [SV79], but is due to Garsia, Rodemich,

Rumsey [GRR71].

Lemma 6.1.2. Let p and Ψ be continuous, strictly increasing functions on [0,∞) such

that p(0) = Ψ(0) = 0 and limt→∞ Ψ(t) = ∞. Given T > 0 and Xt ∈ C([0, T ],R), if

∫ T

0

∫ T

0

Ψ

( |Xt −Xs|p(|t− s|)

)ds dt ≤ B (6.1.1)

then for 0 ≤ s < t ≤ T ,

|Xt −Xs| ≤ 8

∫ (t−s)

0

Ψ−1

(4B

u2

)dp(u). (6.1.2)

In particular,

|Xt −Xs| ≤ 8

∫ (t−s)

0

Ψ−1

(4

u2

∫ T

0

∫ T

0

Ψ

( |Xt −Xs|p(|t− s|)

)ds dt

)dp(u). (6.1.3)

Proof. Define d−1 = T and

I(t) =

∫ T

0

Ψ

( |Xt −Xs|p(|t− s|)

)ds.

Since∫ T

0I(t)dt ≤ B, ∃ t0 ∈ (0, d−1) such that I(t0) ≤ B/T . We may now choose a

non-increasing sequence tn : n ≥ 1 ⊆ [0, t0] in the following manner. Given tn−1, define

dn−1 by p(dn−1) =1

2p(tn−1). Choose tn ∈ (0, dn−1) so that

I(tn) ≤ 2B/dn−1, (6.1.4)

and

Ψ

(∣∣Xtn −Xtn−1

∣∣p(|tn − tn−1|)

)≤ 2I(tn−1)/dn−1. (6.1.5)

Note that if A1 = t ∈ (0, dn−1)|I(t) > 2B/dn−1, then |A1| < dn−1/2. Similarly, if

A2 = t ∈ (0, dn−1)|Ψ(|Xtn−Xtn−1|p(|tn−tn−1|)

)> 2I(tn−1)/dn−1, then |A2| < dn−1/2. Hence,

there must be a point in (0, dn−1) at which both 6.1.4 and 6.1.5 hold. By construction,

2p(dn) = p(tn) ≤ p(dn−1) =1

2p(tn−1).


Therefore, dn ≤ tn ≤ dn−1 ≤ tn−1 and tn → 0 as n →∞. Also,

p(|tn − tn−1|) ≤ p(tn−1) = 2p(dn−1) ≤ 4(p(dn−1)− p(dn)). (6.1.6)

Together, 6.1.4, 6.1.5, 6.1.6 imply

∣∣Xtn −Xtn−1

∣∣ ≤ Ψ−1(2I(tn−1)/dn−1)p(|tn − tn−1|)

≤ Ψ−1(4B/d2n−1)4(p(dn−1)− p(dn))

≤ 4

∫ dn−1

dn

Ψ−1(4B/u2)dp(u).

Now summing over n ≥ 0,

|Xt0 −X0| ≤ 4

∫ T

0

Ψ−1(4B/u2)dp(u). (6.1.7)

By replacing Xt by XT−t and repeating the previous argument we get

|XT −Xt0| ≤ 4

∫ T

0

Ψ−1(4B/u2)dp(u). (6.1.8)

Taken together,

|XT −X0| ≤ 8

∫ T

0

Ψ−1(4B/u2)dp(u). (6.1.9)

This holds for any Ψ, p, and X such that 6.1.1 is satisfied. Thus, given 0 ≤ s ≤ t ≤ T ,

define X(u) = X(s + t−s

Tu), u ∈ [0, T ] and p(u) = p

(t−sT

u), u ∈ [0, T ]. Then through a

natural change of variables

∫ T

0

∫ T

0

Ψ

∣∣∣Xu − Xv

∣∣∣p(|u− v|)

du dv =

(T

t− s

)2 ∫ t

s

∫ t

s

Ψ

( |Xu −Xv|p(|u− v|)

)du dv

≤(

T

t− s

)2

B ≡ B.

Now, replacing X, p, and B in 6.1.9 by X, p, and B respectively we obtain

∣∣∣XT − X0

∣∣∣ ≤ 8

∫ T

0

Ψ−1(4B/u2)dp(u).

An appropriate change of variables produces the desired result.


The following corollaries to the lemma of Garsia, Rodemich and Rumsey, along with

the proofs, are drawn from [QJS02].

Corollary 6.1.3. Let PN be the probability measures on C([0, T ],R) satisfying

EPN[(Xt −Xs)

2] ≤ C(N) |t− s|1+γ (6.1.10)

for t,s ∈ [0, T ] with C(N) ≤ C ≤ ∞ and independent of t and s, and γ > 0. Then

PN are tight. If C(N) → 0 as N → ∞, then PN converge weakly to the trivial process

concentrated on Xt ≡ X0.

Proof. Let Ψ(x) = x2 and apply the lemma of Garsia, Rodemich, and Rumsey for t, s

s.t. |t− s| ≤ δ, then

|Xt −Xs| ≤ 8

∫ δ

0

(4

u2

∫ T

0

∫ T

0

( |Xt −Xs|p(|t− s|)

)2

ds dt

) 12

dp(u).

Hence,

sup|t−s|≤δ

|Xt −Xs| ≤ 8

∫ δ

0

(4

u2

∫ T

0

∫ T

0

( |Xt −Xs|p(|t− s|)

)2

ds dt

) 12

dp(u),

and by Schwarz’s inequality,

EPN

[sup|t−s|≤δ

|Xt −Xs|]≤ 16

∫ δ

0

1

u

(∫ T

0

∫ T

0

EPN

[|Xt −Xs|2]

p(|t− s|)2ds dt

) 12

dp(u).

Now choose p(x) = xα, where 1 + γ2

> α > 1. Using (6.1.10) we have that the right hand

side is bounded above by

16

∫ δ

0

αuα−2

(∫ T

0

∫ T

0

C(N) |t− s|1+γ

|t− s|2α ds dt

) 12

du.

Hence,

EPN

[sup|t−s|≤δ

|Xt −Xs|]≤ A

√C(N)δα−1,

for some A < ∞. This proves that PN are tight. If C(N) → 0 as N →∞ then Xt ≡ X0

is the only possible limit.


Corollary 6.1.4. Let d ≤ 2. Let U be a bounded local function with

U(ρ) = U ′(ρ) = 0.

Let PN be the distribution of XNt =

1

a(N, d)

∫ Nt

0

U(ηs)ds, t ∈ [0, T ] under Pρ. Then PN

converge weakly to the trivial process concentrated on Xt ≡ 0.

Proof. By Fubini’s theorem

Eνρ

[(∫ t

0

U(ηs)ds

)2]

=

∫ t

0

∫ t

0

Eνρ [U(ηs)U(ηr)] ds dr

= 2

∫ t

0

∫ s

0

Eνρ [U(ηs)U(ηr)] dr ds

= 2

∫ t

0

∫ s

0

Eνρ [U(ηs−r)U(η0)] dr ds

= 2

∫ t

0

∫ s

0

Eνρ

[U(η0)Eνρ [U(ηs−r)|η0]

]dr ds

= 2

∫ t

0

∫ s

0

Eνρ [U(η0)Ps−rU(η0)] dr ds

= 2

∫ t

0

∫ s

0

Eνρ

[(P s−r

2U(η0)

)2]

dr ds.

Employing the decay estimate, Theorem 5.3.3, the last line is equal to

a2(t, d)C1[U′(ρ)]2χ(ρ)([8πΦ′(ρ)])−d/2 + C2t

2−d/2−γ,

in d = 1, 2, where C1, C2 are constants depending upon d and ρ such that C1, C2 < ∞.

Since U ′(ρ) = 0, this is equal to C2t2−d/2−γ. As the process is stationary,

EPN[(XN

t −XNs )2] = C(N) |t− s|2−d/2−γ ,

where C(N) = O(Nα), where α > 0. Thus, the conditions of Corollary 6.1.3 are satisfied

and XNt converges to the trivial process as desired.


6.2 Central Limit Theorem: Specific Case

We will make use of the following construction, which may be found in Quastel, Jankowski,

and Sheriff [QJS02]. Consider the symmetric simple random walk on Zd. Define,

pt(x, y) = e−2dt

∞∑n=0

(2dt)n

n!pn(x, y),

where pn(x, y) are the n-step transition probabilities for such a random walk. Then

pt(x, y) is the solution of

∂p

∂t= ∆p, p0(x, y) = δx(y)

where ∆ represents the lattice Laplacian, as defined earlier (see 2.0.1). Now let ut(x) =∫ Φ′(ρ)t

0ps(0, x)ds and note that ut(x) is the solution of

∂u

∂t= Φ′(ρ)[∆u + δ0], u0(x) ≡ 0.

Finally, letting

UTt (η) =

∑

x∈Zd

uT−t(x)(η(x)− ρ),

produces a function U such that (∂ + L)U = V ′(η(0)) − Φ(ρ) plus terms that are man-

ageable as desired. Then MTt = UT

t (ηt) − UT0 (η0) −

∫ t

0(∂s + L)UT

s (ηs)ds is a martingale


in t up to time T . Now

(∂s + L)UTs (ηs)

=∑

x∈Zd

(−Φ′(ρ))pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ)

−1

2

∑

x∈Zd

uT−s(x)∑y∼z

(V ′(η(z))− V ′(η(y)))(∂η(z) − ∂η(y))ηs(x)

= −∑

x∈Zd

(−Φ′(ρ))pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ)

+∑

x∈Zd

uT−s(x)∑y∼x

(V ′(ηs(y))− V ′(ηs(x)))

= −∑

x∈Zd

(−Φ′(ρ))pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ) +∑

x∈Zd

uT−s(x)∆V ′(ηs(x))

= −∑

x∈Zd

(−Φ′(ρ))pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ) +∑

x∈Zd

V ′(ηs(x))∆uT−s(x)

= −∑

x∈Zd

(−Φ′(ρ))pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ)

+∑

x∈Zd

V ′(ηs(x))

(1

Φ′(ρ)∂suT−s(x) + δ0(x)

)

= −∑

x∈Zd

(−Φ′(ρ))pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ) +∑

x∈Zd

V ′(ηs(x))(−pΦ′(ρ)(T−s)(0, x)

)

+V ′(ηs(0))− Φ(ρ) + Φ(ρ).

Therefore,

MTT = −UT

0 +

∫ T

0

∑

x∈Zd

Φ′(ρ)pΦ′(ρ)(T−s)(0, x)(ηs(x)− ρ)ds

−∫ T

0

∑

x∈Zd

pΦ′(ρ)(T−s)(0, x)(V ′(ηs(x))− Φ(ρ))ds

−∫ T

0

(V ′(ηs(0))− Φ(ρ))ds.

Hence,∫ NT

0(V ′(ηs(0))− Φ(ρ))ds = −MNT

NT − UNT0 + RNT , where

RNT =

∫ NT

0

∑

x∈Zd

[Φ′(ρ)(ηs(x)− ρ)− (V ′(ηs(x))− Φ(ρ))] pΦ′(ρ)(NT−s)(0, x)ds.

For notational convenience we will set b(ηs(x)) = Φ′(ρ)(ηs(x) − ρ) − (V ′(ηs(x)) − Φ(ρ))

and qs(x) = pΦ′(ρ)(s)(0, x).


Letting XNt =

1

a(N, d)

∫ Nt

0

(V ′(ηs(0))− Φ(ρ))ds, by using the decay estimate and a

variance calculation as in Corollary 6.1.4, we can see that Corollary 6.1.3 is satisfied, for

some C(N) < ∞, which shows the XNt are tight. Note also that

Eνρ [b(ηt(x))] = 0 and b′(ρ) = ∂αEνα [b(ηt(x))]|α=ρ = 0. Also∑

x∈Zd

qt(x) = 1.

In order to prove the desired result we will first show that the remainder term

1a(N,d)

RNT is negligible in the limit. Note that it is mean zero and that the variance

may be written

1

a(N, d)2

∫ NT

0

∫ NT

0

∑

x,y∈Zd

qNT−r(x)qNT−s(y)Eνρ [b(ηr(x))b(ηs(y))]dr ds

Since the invariant measures are reversible,

Eνρ [b(ηr(x))b(ηs(y))]

= Eνρ

[b(η0(x))b(η|r−s|(y))

]

= Eνρ

[b(η0(x))Eνρ

[b(η|r−s|(y))|η0

]]

= Eνρ

[b(η0(x))P|r−s|b(η(y))

]

= Eνρ

[P |r−s|

2

b(η(x))P |r−s|2

b(η(y))]

≤ Eνρ

[(P |r−s|

2

b(η(x)))2

]+ Eνρ

[(P |r−s|

2

b(η(y)))2

].

By our earlier Theorem 3.3 regarding L2 decay

∣∣∣td/2Eνρ

[(Ptb(η))2]− C(ρ, d)

(b′(ρ)

)2∣∣∣ ≤ C1t

−γ,

for γ > 0, C1 < ∞. Since b′(ρ) equals zero, we have that

Eνρ

[(Ptb(η(x)))2] ≤ C1t

− d2−γ.

Hence, Eνρ [b(ηr(x))b(ηs(y))] ≤ C1

(|r − s|− d

2−γ

)uniformly in x and y. Thus, since


∑x∈Zd qt(x) = 1 for all t and for all x, we have

1

a(N, d)2(RNT )2 ≤ 1

a(N, d)2

∫ NT

0

∫ NT

0

|r − s|− d2−γ drds

=2

a(N, d)2

∫ NT

0

∫ s

0

|s− r|− d2−γ drds

=C(d, γ)

a(N, d)2(NT )2− d

2−γ,

where C(d, γ) < ∞. Recalling that a(N, d) = N3/4 for d = 1 and a(N, d) =√

N log N

for d = 2 we can see that in dimensions one and two, the remainder term goes to the

constant 0 in the limit as N → 0.

Now,

dU ts(ηs) = ∂sU

tsds +

∑

x∈Zd

∂ηsUts(ηs)dηs.

Since,

dηs(x) = ∆V ′(ηs(x))ds +∇dBs(x),

we have that

dU ts(ηs) = ∂sU

tsds +

∑

x∈Zd

ut−s(x)∆V ′(ηs(x))ds +√

2∑

x∈Zd

ut−s(x)∇dBs(x)

= (∂s + L) U ts(ηs)ds +

√2

∑

x∈Zd

ut−s(x)∇dBs(x).

Therefore,

MTt = UT

t (ηt)− UT0 (η0)−

∫ t

0

(∂s + L) U ts(ηs)ds

=√

2

∫ t

0

∑

x∈Zd

uT−s(x)∇dBs(x).

Hence,

MNTNT =

√2

∑

x∈Zd

∫ NT

0

uNT−s(x)∇dBs(x)

=√

2∑

x∈Zd

∫ NT

0

∇uNT−s(x)dBs(x),


and is therefore a Gaussian process.

Finally, recall that

UNT0 =

∑

x∈Zd

uNT (x)(η0(x)− ρ).

An appeal to the central limit theorem identifies UNT0 as being a Gaussian process, while

the L2 decay establishes that UNT0 and MNT

NT are asymptotically independent. Thus, in

the limit, 1a(NT,d)2

∫ NT

0(V ′(ηs(0)) − Φ(ρ))ds is a Gaussian process, while the covariance

structure is determined through the L2 decay. Of interest is the behaviour in one dimen-

sion, where the process converges to fractional Brownian motion, and the behaviour in

higher dimensions, where Brownian motion is the limiting process.

6.3 CLT for general functions

The proof of Theorem 3.3.2, the invariance principle for mean-zero bounded local func-

tions, proceeds as follows.

Proof. Proof of Theorem 3.3.2

Given g, a mean-zero bounded local function, write

1

a(N, d)

∫ Nt

0

g(ηs)ds =1

a(N, d)

∫ Nt

0

(g(ηs)− g′(ρ)

Φ′(ρ)[V ′(ηs(0))− Φ(ρ)]

)ds

+1

a(N, d)

g′(ρ)

Φ′(ρ)

∫ Nt

0

[V ′(ηs(0))− Φ(ρ)] ds.

By corollary 6.1.4 the first term converges to the trivial process Xt ≡ 0, while Lemma

6.1.1 previously established that the second term converges to the desired limit.

Chapter 7

Future Work

7.1 Central Limit Theorem for Ginzburg-Landau

The proof of the decay to equilibrium and central limit theorem for the Ginzburg-Landau

model, relied upon a number of assumptions, regarding the potential V and the function

under consideration g. For example, we have assumed that our function g is both bounded

and has finite triple norm. It would be natural to consider relaxing each assumptions.

Also, we have assumed the potential V to be a slightly perturbed version of a quadratic

potential and one could consider relaxing this assumption. Finally, in proving the rate

of decay, our first step was to establish the decay result for a special function. It would

be of interest to determine if a simpler function is available that will permit the proof of

the desired decay result. If successful, this might suggest an approach for other models.

7.2 Central Limit Theorem for ∇− φ Interface

Model

A model closely related to the Ginzburg-Landau model is the ∇ − φ interface model.

This is a model of the interface that may exist between two pure thermodynamic phases,

89

Chapter 7. Future Work 90

at low temperatures.

At each site x in the d-dimensional lattice Zd, ϕ(x) is a real valued random vari-

able representing the height of the interface at site x, with neighbouring heights related

through some potential V .

Denote by eα, α = 1, . . . , d, the unit vector in the direction α, i.e. (eα)β = δαβ. Zd∗ is

the set of positively directed bonds, b = (x, x + eα), for some x and some α. Bonds are

denoted by b = (x, y) = (xb, yb) and −b = (y, x) = (yb, xb).

The objects of interest are the differences in heights, η(b) = η(x, y) = ϕ(y) − ϕ(x),

where b = (x, y). Let η(x, α) = η(b) for b = (x, x + α) and also define η(−b) = −η(b).

Given a function f : Zd → R, set ∇αf(x) = f(x+ eα)−f(x), ∇∗αf(x) = f(x− eα)−f(x)

and ∆1 =∑d

α=1∇α∇∗α.

For each height configuration, ϕ, there is a unique increment configuration, η with

the property that∑

b∈Cη(b) = 0,

for every closed loop C. This is known as the plaquette condition. Let χ be the subset

of RZd∗satisfying the plaquette condition.

The gradient field η behaves according to the SDE’s

dηt(x, α) = −d∑

β=1

∇α∇∗βV ′(η(x, β))dt +

√2∇αdBt(x).

Hydrodynamic behaviour and equilibrium fluctuations have been investigated by Fu-

naki and Spohn [FS97] and Giacomin, Olla and Spohn [GOS01]. In d = 1 the ∇ − φ

interface model reduces to the Ginzburg-Landau model. Thus the decay result and cen-

tral limit theorem from previous chapters carries over. However, one would wish to

establish both results in for d ≥ 2.

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