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.
Chapter 1. Introduction 3
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
Chapter 1. Introduction 4
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
Chapter 1. Introduction 5
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
Chapter 3. Ginzburg-Landau Process 11
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).
Chapter 3. Ginzburg-Landau Process 12
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
Chapter 3. Ginzburg-Landau Process 13
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
Chapter 3. Ginzburg-Landau Process 14
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
Chapter 3. Ginzburg-Landau Process 15
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)
where χ(ρ) = Eνρ [(η0 − ρ)2], Φ(ρ) = Eνρ [V′(η(0))], and Φ′(ρ) = ∂αEνα [V ′(η(0))]|α=ρ.
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
Chapter 3. Ginzburg-Landau Process 16
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
⟩
Chapter 4. Approximation Results: Ginzburg-Landau 19
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.
Chapter 4. Approximation Results: Ginzburg-Landau 20
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
Chapter 4. Approximation Results: Ginzburg-Landau 21
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).
Chapter 4. Approximation Results: Ginzburg-Landau 22
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).
Chapter 4. Approximation Results: Ginzburg-Landau 23
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
Chapter 4. Approximation Results: Ginzburg-Landau 24
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.
Chapter 4. Approximation Results: Ginzburg-Landau 25
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].
Chapter 4. Approximation Results: Ginzburg-Landau 26
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),
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 29
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),
where χ(ρ) = Eνρ [(η0 − ρ)2], Φ(ρ) = Eνρ [V′(η(0))], and Φ′(ρ) = ∂αEνα [V ′(η(0))]|α=ρ.
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 30
Lemma 5.2.1.
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),
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
J(t, x)τx Avgy∈Λk
(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),
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 31
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)
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 32
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
(∆J)(t− r, x)τx Avgy∈Λk
(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
(∆J)(t− r, x)τx Avgy∈Λk
(V ′(ηr(y))−Φ(ρ)−Φ′(ρ) (ηr(y)− ρ)
)dr
])2].
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 33
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 .
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 34
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 .
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 35
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 36
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 .
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 37
Integrating by parts,
∫ t
0
1
|ΛK |∑
x∈Zd
(∆J) (t− r, x)τx
Avgy∈Λk
[V ′(ηr(y))−GkV
′(ηr(0))]
dr
=1
|ΛK |∑
x∈Zd
(∆J) (0, x)
∫ t
0
τx
Avgy∈Λk
[V ′(ηr(y))−GkV
′(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
[V ′(ηr(y))−GkV
′(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
[V ′(ηr(y))−GkV
′(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
[V ′(ηr(y))−GkV
′(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] .
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 38
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 .
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 39
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))
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 40
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
(∆J)(t− r, x)τx Avgy∈Λk
(ηr(y)− ρ) dr
+
∫ s
0
1
|ΛK |∑
x∈Zd
J(t− r, x)τx Avgy∈Λk
L (ηr(y)− ρ) dr.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 41
A straightforward calculation shows that
L (ηr(y)− ρ) = ∆V ′(ηr(y)).
We further note that
1
|ΛK |∑
x∈Zd
J(t− r, x)τx Avgy∈Λk
L (ηr(y)− ρ)
=1
|ΛK |∑
x∈Zd
J(t− r, x)τx Avgy∈Λk
∆V ′(ηr(y))
=1
|ΛK |∑
x∈Zd
(∆J)(t− r, x)τx Avgy∈Λk
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
(∆J)(t− r, x)τx Avgy∈Λk
[V ′(ηr(y))− Φ(ρ)− Φ′(ρ) (ηr(y)− ρ)] dr
=
∫ s
0
1
|ΛK |∑
x∈Zd
(∆J)(t− r, x)τx Avgy∈Λk
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
J(t, x)τx Avgy∈Λk
(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
(∆J)(t− r, x)τx Avgy∈Λk
W (ηr(y))dr
]
=1
|ΛK |∑
x∈Zd
J(t, x)τx Avgy∈Λk
(η(y)− ρ).
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 42
Hence,
Eη
[1
|ΛK |∑
x∈Zd
J( x
K
)τx Avg
y∈Λk
(ηt(y)− ρ)
]
=1
|ΛK |∑
x∈Zd
J(t, x)τx Avgy∈Λk
(η(y)− ρ)
+ Eη
[∫ t
0
1
|ΛK |∑
x∈Zd
(∆J)(t− r, x)τx Avgy∈Λk
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
J(t, x)τx Avgy∈Λk
(η(y)− ρ)
)2
+ (1 + A)Eνρ
[(Eη
[∫ t
0
1
|ΛK |∑
x∈Zd
(∆J)(t− r, x)τx Avgy∈Λk
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
J(t, x)τx Avgy∈Λk
(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
J(t, x)τx Avgy∈Λk
(η(y)− ρ)
)2
=χ(ρ)
[8πΦ′(ρ)]d/2.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 43
We may similarly rewrite (5.2.16) to show that
Eνρ
(1
|ΛK |∑
x∈Zd
J(t, x)τx Avgy∈Λk
(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
(∆J)(t− r, x)τx Avgy∈Λk
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 44
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
(∆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. (5.3.2)
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 45
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).
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 46
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,
for some constant C and α > 0.
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)
for some finite constant C and α > 0.
Proof. For t ≥ 0, let gt stand for Ptg. Thus, gt is the solution of the backward equation
∂tgt = Lgt
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 47
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,
for some constant C and α > 0.
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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 48
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,
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 49
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 50
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 51
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
s(d+2)/2D(νρ, τX(s)vs)ds +
(d + 2
2
) ∫ t
to
sd/2⟨(τX(s)vs)
2⟩ds
= −2
∫ t
to
s(d+2)/2D(νρ, τX(s)vs)ds +
(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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 52
≤ −2
∫ t
to
s(d+2)/2D(νρ, τX(s)vs)ds + C (ρ, d) log
(t
t0
) ⟨g2
⟩ds
+
∫ t
to
s(d+2)/2D(νρ, τX(s)vs)ds +
(d + 2
2
) ∫ t
to
sd/2⟨(Gl,LτX(s)vs)
2⟩ds
≤(
d + 2
2
) ∫ t
to
sd/2⟨(Gl,LτX(s)vs)
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
.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 53
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
⟨(Gl,Lτx+y (gs − (Gkg)s))
2⟩]]
ds ≤ Ct1−α, (5.3.14)
for some finite constant C and α > 0.
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
⟨(Gl,L (τxgs − τx+ygs))
2⟩]]
ds ≤ Ct1−α,
for some finite constant C and α > 0.
Lemma 5.3.5.∫ t
to
Avgx∈Λl/4
[Avgy∈ΛK
[J
( y
K
)sd/2
⟨(Gl,Lτx+y (gs − (Gkg)s))
2⟩]]
ds ≤ Ct1−α,
for some finite constant C and α > 0.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 54
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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 55
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)
⟩.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 56
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〉 .
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 57
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]
.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 58
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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 59
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)〉,
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 60
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).
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 61
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)
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 62
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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 63
(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)
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 64
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
].
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 65
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).
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 66
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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 67
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
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 68
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
]
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 69
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,
Eνρ [H(ft)] ≤ C2
√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
)d (log l + log δ−1
)
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 70
Since l ≥ 1 and δ ≤ 1, we have that t∗ ≤ 1, and since H(ft) decreases in time,
Eνρ [H(ft)] ≤ C2
(L
l
)d (log l + log δ−1
),
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).
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 71
Lemma (5.3.4).
∫ t
to
Avgx∈Λl/4
[Avgy∈ΛK
[J
( y
K
)sd/2
⟨(Gl,L (τxgs − τx+ygs))
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),
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 72
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
)d (log l + log δ−1
).
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 73
Therefore, the previous term is bounded above by
t1−εC(d, g, ρ)
(L
l
)d (log l + log δ−1
).
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
⟨(Gl,Lτx+y (gs − (Gkg)s))
2⟩]]
ds ≤ Ct1−α,
for some finite constant C, independent of γ, and α > 0.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 74
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
]
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 75
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
[(g −Gkg)τ−(x+y)fs
]is bounded above by 2‖g‖∞, we
have that
Eνρ
[(g −Gkg)τ−(x+y)fs
] ≤ 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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 76
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
)d (log l + log δ−1
).
Therefore, the previous term is bounded above by
t4εA(J)C(d, g, ρ)
(L
l
)d (log l + log δ−1
)
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.
Chapter 5. Relaxation to Equilibrium: Ginzburg-Landau 77
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)
where χ(ρ) = Eνρ [(η0 − ρ)2], Φ(ρ) = Eνρ [V′(η(0))], and Φ′(ρ) = ∂αEνα [V ′(η(0))]|α=ρ.
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.
Chapter 6. Central Limit Theorem: Ginzburg-Landau 80
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).
Chapter 6. Central Limit Theorem: Ginzburg-Landau 81
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.
Chapter 6. Central Limit Theorem: Ginzburg-Landau 82
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.
Chapter 6. Central Limit Theorem: Ginzburg-Landau 83
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.
Chapter 6. Central Limit Theorem: Ginzburg-Landau 84
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
Chapter 6. Central Limit Theorem: Ginzburg-Landau 85
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).
Chapter 6. Central Limit Theorem: Ginzburg-Landau 86
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
Chapter 6. Central Limit Theorem: Ginzburg-Landau 87
∑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),
Chapter 6. Central Limit Theorem: Ginzburg-Landau 88
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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