Diffusion of Colour in the Simple Exclusion Process

Jeremy Quastel University of California, Davis

Abstract

We prove a diffusion scaling limit for the macroscopic densities of coloured particles performing the simply excluded random walk, and relate this to the limiting behaviour of a test particle in equilibrium.

1. Introduction

In a large system of interacting particles it is often of interest to study the asymptotic motion of one or more tagged particles in the system. One of the simplest ways to approach this is to consider a system of coloured, but mechanically identical, particles.

Lebowitz and Spohn have shown [L–S] that if, for the system in equilibrium, a rescaled test particle converges to Brownian motion (self–diffusion), then the nonequilibrium system has a hydrodynamical limit. Furthermore, if several test particles converge to independent Brownian motions (propagation of chaos), then colour density profiles have the hydrodynamic limit.

Both properties have been verified for the symmetric simple exclusion process in equilibrium by Kipnis and Varadhan [K–V]. In particular, their results imply that when the system, disregarding colour, is in equlibrium, the colour density profiles evolve asymptotically independently according to a linear diffusion equation.

In this paper we again consider the symmetric simple exclusion process, but not necessarily in equilibrium. Instead of proving propagation of chaos, we prove directly the hydrodynamical limit for the evolving colour profiles. These are shown to evolve asymptotically according to a coupled nonlinear diffusion equation. The method we use is an adaptation of the method developed by Varadhan for the non–gradient Ginzburg–Landau model [V2], which, in turn, is a refinement of the entropy method of Guo, Papanicolaou and Varadhan [G–P–V].

The model has two characteristics which conspire to make the proof of the hydrodynamic limit difficult. First, the system is of non–gradient type. In this situation the usual idea of replacing microscopic information by averages with respect to a slowly varying local equilibrium does not suffice. Lower order perturbations actually contribute to the limit, and one has to somehow show that the correct state is the one which minimises the local entropy production. In particular, we found it necessary to prove the spectral gap for the process. This estimate is in Section .

Second, the generator of the process is degenerate in the sense that considered as a speed change model the rates often vanish. Because of this, many of the standard estimates involving the Dirichlet form (see [G–P–V] or [K–O–V]) are not available. We get around this problem by using a kind of penalisation. These estimates are contained in Sections and .

For simplicity, the theorem has been stated and proved for the system with only two colours. The generalisation to any finite number of colours is immediate.

I would like to take this opportunity to thank S.R.S. Varadhan for all his help and support. Also, some ideas of F. Rezakhanlou considerable simplified the original proof of Theorem 0.7. His help is gratefully acknowledged.

2. Notation and results

The symmetric simple exclusion process is described as follows: For each positive integer N, let T_{N} be the periodic lattice {(1, …, N)^{d}, with addition modulo N}. We think of \frac{1}{N} T_{N} = \{\frac{u}{N} : u ∈ T_{N}\} as embedded in T^{d}, the standard d–torus. Particles are distributed on T_{N} and perform a simply excluded random walk: Each particle is held for an exponential (mean 1/N^{2}) time, then chooses a nearby site according to a symmetric law p(v) with finite range. If the site is empty the particle jumps there; if not, the jump is suppressed. We assume, for simplicity, that p > 0 for the jump of length 1 in any direction.

In this paper we consider the process in which two species of mechanically identical particles are distinguished, say blue particles and white particles. A configuration can be represented by η ∈ X_{N} = {−1, 0, 1}^{T_{N}}, where

η(u) = \{ \begin{matrix}1 & if there is a blue particle at u \\ 0 & if there is no particle at u \\ −1 & if there is a white particle at u \end{matrix}

or by the two functions β(u) = 1_{η(u)=1} and ω(u) = 1_{η(u)=−1}. The system is a Markov process with state space X_{N} and generator

L_{N}f = N^{2} ∑_{u,v∈T_{N}} p(v)a(u, u + v)D^{u,u+v}f(η)

where a(u, u + v) = 1_{η(u)η(u+v)=0} represents the exclusion and

D^{u,u+v}f(η) = f(η^{u,u+v}) − f(η)

where

η^{u,u+v}(x) = \{ \begin{matrix}η(x) & if x ≠ u or u + v \\ η(u + v) & if x = u \\ η(u) & if x = u + v. \end{matrix}

Let π^{N} be the measure on X_{N} which gives mass \frac{1}{3^{N^{d}}} to each configuration. L_{N} is symmetric with respect to π^{N} and the Dirichlet form is given by

D_{N}(f) = −\frac{1}{N^{2}} Eπ^{N} [fL_{N}f] = \frac{1}{2}Eπ^{N} [ ∑_{u,u+v} p(v)a(u, u + v)(D^{u,u+v}f)^{2}].

Let δ_{u/N} be the measure giving mass 1 to the point \frac{u}{N} ∈ T^{d}, and define profile measures,

μβ^{N}(η) = \frac{1}{N^{d}} ∑_{u∈T_{N}} β(u) · δ_{u/N} and μω^{N}(η) = \frac{1}{N^{d}} ∑_{u∈T_{N}} ω(u) · δ_{u/N}.

Definition 1. A sequence of densities f_{N} is said to have macroscopic profile ρ_{β}(θ), ρ_{ω}(θ) if for all δ > 0 and smooth test functions ϕ and ψ,

lim_{N→∞} P^{f_{N}π^{N}} [ | ∫_{T^{d}} ϕ(θ)μβ^{N}(dθ) − ∫_{T^{d}} ϕ(θ)ρ_{β}(θ)dθ |

+ | ∫_{T^{d}} ψ(θ)μω^{N}(dθ) − ∫_{T^{d}} ψ(θ)ρ_{ω}(θ)dθ |> δ] = 0

and we write f_{N} ∼ ρ_{β}(θ), ρ_{ω}(θ).

The main result is that the colour density profiles have a hydrodynamic limit.

Theorem 2.1. Suppose the process has initial distribution f_{N}^{0}π^{N} where

f_{N}^{0} ∼ ρ_{β}^{0}(θ), ρ_{ω}^{0}(θ)

where ρ_{β}^{0} + ρ_{ω}^{0} < 1. Then at a later time t it has distribution f_{N}^{t}π^{N} with f_{N}^{t} ∼ ρ_{β}(t, θ), ρ_{ω}(t, θ) where ρ_{β}(t, θ), ρ_{ω}(t, θ) is the solution of

\frac{∂}{∂t} ( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} ) = \frac{1}{2}∇·( \begin{matrix}\frac{ρ_{ω}}{ρ_{β}+ρ_{ω}} D_{s} + \frac{ρ_{β}}{ρ_{β}+ρ_{ω}} D & \frac{ρ_{β}}{ρ_{β}+ρ_{ω}} (D − D_{s}) \\ \frac{ρ_{ω}}{ρ_{β}+ρ_{ω}} (D − D_{s}) & \frac{ρ_{β}}{ρ_{β}+ρ_{ω}} D_{s} + \frac{ρ_{ω}}{ρ_{β}+ρ_{ω}} D \end{matrix} )∇( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} )
ρ_{β}(0, θ) = ρ_{β}^{0}(θ)
ρ_{ω}(0, θ) = ρ_{ω}^{0}(θ).  (2.1)

Here D is the d × d matrix with entries D_{ij} = 2 ∑_{v} p(v)v_{i}v_{j}, and D_{s} = D_{s}(ρ_{β} + ρ_{ω}) is the self–diffusion covariance. Both are described below.

Define the colourblind macroscopic profile ρ = ρ_{β} + ρ_{ω}. Under the assumptions of the theorem, it is well known that ρ(t, θ) evolves according to the heat equation,

\frac{∂ρ}{∂t} = \frac{1}{2}∇ · D∇ ρ,  ρ(0, θ) = ρ^{0}(θ).  (2.2)

See, for example, [K–O–V].

To describe D_{s} we construct a new process: On Z^{d} infinitely many particles perform the simply excluded random walk. The particles are indistinguishable except for one test particle. If the state of the system is described by the position x of the test particle, and the environment ζ ∈ {0, 1}^{Z^{d}−{0}} as seen from the test particle, the generator of the (x, ζ) process is given by

L̃f(x, ζ) = 2 ∑_{v≠0} p(v)(1 − ζ(v))(f(x + v, τ_{−v}ζ) − f(x, ζ))

+ ∑_{u,u+v≠0} p(v)(f(x, ζ^{u,u+v}) − f(x, ζ)).

The ζ process alone is reversible with respect to π_{ρ}, the product measure with constant density ρ. Let P_{ρ}^{0} be the measure generated by starting x at 0 and the ζ process in π_{ρ}. Kipnis and Varadhan have proved the following central limit theorem for the test particle.

Theorem 2.2. The rescaled test particle, \frac{1}{N} x(N^{2}t), converges in the Skorohod space to a diffusion with covariance

(l, D_{s}(ρ)l) = lim_{t→∞} \frac{E^{P_{ρ}^{0}}(l · x(t))^{2}}{t} .

Based on this we can give a heuristic derivation of Theorem 0.1: For large N a particle at θ acts as if it were a test particle in Z^{d} with the other particles distributed according to π_{ρ(θ)}. In addition, it is subject to a drift, b, caused by the evolution of the entire system towards equilibrium. Thus in the limit one expects it to converge to a diffusion with generator

L = \frac{1}{2}D_{s}∇ · ∇ + b · ∇.

Macroscopic profiles should evolve according to the forward equations

\frac{∂ρ}{∂t} = L^{∗}ρ  (2.3)
\frac{∂ρ_{β}}{∂t} = L^{∗}ρ_{β}  (2.4)
\frac{∂ρ_{ω}}{∂t} = L^{∗}ρ_{ω}.  (2.5)

Since ρ also satisfies the heat equation (0.2), (0.3) gives

\frac{1}{2}∇ · ∇D_{s}ρ − ∇ · bρ = \frac{1}{2}∇ · D∇ρ,

which forces

b = \frac{1}{2} \frac{∇((D_{s} − D)ρ)}{ρ}.

Then (0.4) and (0.5) become

\frac{∂ρ_{β}}{∂t} = \frac{1}{2}∇ · (∇(D_{s}ρ_{β}) − \frac{∇((D_{s} − D)ρ)}{ρ}ρ_{β})  (2.6)
\frac{∂ρ_{ω}}{∂t} = \frac{1}{2}∇ · (∇(D_{s}ρ_{ω}) − \frac{∇((D_{s} − D)ρ)}{ρ}ρ_{ω}),  (2.7)

which is equivalent to (0.1). Notice that if ρ is constant then these equations are linear:

\frac{∂}{∂t} ( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} ) = \frac{1}{2}∇·( \begin{matrix}D_{s}(ρ) & 0 \\ 0 & D_{s}(ρ) \end{matrix} )∇( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} ) .

Remark on ergodicity. Consider the “hyperplane” of ρ_{β}N blue and ρ_{ω}N white particles, where ρ_{β}, ρ_{ω}, ρ_{β} + ρ_{ω} ∈ [0, 1] and ρ_{β}N, ρ_{ω}N are integers, and let u_{ρ_{β},ρ_{ω}}^{N} denote the uniform measure on this set. The hyperplane is preserved by the process, and it is reversible with respect to u_{ρ_{β},ρ_{ω}}^{N}. If the model is not the one dimensional nearest neighbour, and if ρ_{β} + ρ_{ω} < 1 , then the process is ergodic with respect to u_{ρ_{β},ρ_{ω}}^{N}. For the 1–d nearest neighbour case this is not true. Furthermore, in this case, the self–diffusion constant vanishes. Nevertheless, Theorem 0.1 is true as stated. In this case the heuristic derivation above can easily be made into a rigorous proof. In all the other cases D_{s} > 0, and it is hard to make the heuristic derivation rigorous, so we use a completely different method. We will assume in the following that p(2) > 0 for the 1–d model.

Remark on uniqueness. From the form of the equations (0.6), (0.7) we see that the system of equations is coupled only through the total density ρ = ρ_{β} + ρ_{ω}. Since ρ is evolving according to the heat equation this immediately gives us uniqueness of positive solutions to the system (0.6), (0.7), and therefore to (0.1).

3. Homogenisation formula

Let M(a, b) = Eπ^{N} [exp(aβ(u) + bω(u))]. Then

h(ρ_{β}, ρ_{ω}) = ρ_{β} log ρ_{β} + ρ_{ω} log ρ_{ω} + (1 − ρ_{β} − ρ_{ω}) log(1 − ρ_{β} − ρ_{ω})

is the convex conjugate of log M(a, b) and we have conjugate coordinates

λβ = log \frac{ρ_{β}}{1 − ρ} = \frac{∂h}{∂ρ_{β}} and λω = log \frac{ρ_{ω}}{1 − ρ} = \frac{∂h}{∂ρ_{ω}} .

Then

∇ ( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} ) = (h′′)^{−1}∇ ( \begin{matrix}λβ \\ λω \end{matrix} ) = ( \begin{matrix}ρ_{β}(1 − ρ_{β})I & −ρ_{β}ρ_{ω}I \\ −ρ_{β}ρ_{ω}I & ρ_{ω}(1 − ρ_{ω})I \end{matrix} ) ∇ ( \begin{matrix}λβ \\ λω \end{matrix} ) .

The 2d×2d matrix h′′ is known as the compressibility. In this notation (0.1) becomes

\frac{∂}{∂t} ( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} ) = \frac{1}{2}∇ · A∇ ( \begin{matrix}λβ \\ λω \end{matrix} ) ,

where

A = ( \begin{matrix}\frac{ρ_{β}ρ_{ω}}{ρ} D_{s} + \frac{ρ_{β}^{2}(1−ρ)}{ρ}D & \frac{−ρ_{β}ρ_{ω}}{ρ} D_{s} + \frac{ρ_{β}ρ_{ω}(1−ρ)}{ρ}D \\ \frac{−ρ_{β}ρ_{ω}}{ρ} D_{s} + \frac{ρ_{β}ρ_{ω}(1−ρ)}{ρ}D & \frac{ρ_{β}ρ_{ω}}{ρ} D_{s} + \frac{ρ_{ω}^{2}(1−ρ)}{ρ}D \end{matrix} )

The advantage here is that A is symmetric. The associated quadratic form on R^{d}×R^{d} is

((a, b), A(a, b)) = \frac{ρ_{β}ρ_{ω}}{ρ} ((a − b), D_{s}(a − b)) + ρ(1 − ρ)((\frac{ρ_{β}}{ρ} a + \frac{ρ_{ω}}{ρ} b), D(\frac{ρ_{β}}{ρ} a + \frac{ρ_{ω}}{ρ} b)).

There is a variational formula for D_{s}. For the (x, ζ) process define the Dirichlet form

D̃_{ρ}(f) = E^{dx⊗π_{ρ}}[∑_{v≠0} p(v)(1 − ζ(v))(f(x + v, τ^{−v}ζ) − f(x, ζ))^{2}

+ \frac{1}{2} ∑_{u,u+v≠0} p(v)(f(x, ζ^{u,u+v}) − f(x, ζ))^{2}],

where dx is the measure on Z^{d} which gives each site mass 1. The next theorem is a consequence of results in [K–V]. It also appears in [S1].

Theorem 3.1. The self–diffusion covariance D_{s}(ρ) is given by the variational formula

(l, D_{s}(ρ)l) = 2 inf_{ψ=ψ(ζ)} D̃_{ρ}(l · x + ψ).

The main result of this section is a variational formula for A. We start with some notation which will be used here and later. Let π_{ρ_{β}, ρ_{ω}} be the product measure on X = {−1, 0, 1}^{Z^{d}} with

Eπ_{ρ_{β}, ρ_{ω}} [β(u)] = ρ_{β} and Eπ_{ρ_{β}, ρ_{ω}} [ω(u)] = ρ_{ω}.

A cylinder function is a function on X which depends only on {η(u)}_{u∈F} for some finite set F ⊂ Z^{d}. Let S = {v ∈ R^{d} : p(v) > 0}. For a cylinder function g the function f_{g} : X → R^{S} with coordinates

f_{g}^{v} = D^{0,v} ∑_{u∈Z^{d}} τ^{u}g

is well defined. Any function f : X → R^{S} which can be written f = f_{g} for some cylinder function g will be called exact. Let E_{0} denote the set of exact functions. Define

H = {f : Eπ_{ρ_{β}, ρ_{ω}} [∑_{v} p(v)a(0, v)(f^{v})^{2}] < ∞}  (3.1)

and T ⊂ H to be the closure in H of the linear span of functions of the form ∑_{u}(aβ(u) + bω(u))ψ_{u}^{v}(ζ), where ψ_{u}^{v} are colourblind functions. Let T^{⊥} be the orthogonal complement of T in H, and define the operator D(f) = {D^{0,v}f^{v}}_{v∈S}. The subspace T is preserved by the evolution in the sense that DT ⊂ T . Since D is symmetric we also have DT^{⊥} ⊂ T^{⊥}. For a cylinder function g define Q_{g}(ρ_{β}, ρ_{ω}, a, b) to be the function

Eπ_{ρ_{β}, ρ_{ω}}[ ∑_{v} p(v)a(0, v)(v·a (β(v) − β(0)) + v·b (ω(v) − ω(0)) + f_{g}^{v})^{2}].  (3.2)

Theorem 3.2. The quadratic form ((a, b), A(a, b)) is given by the variational formula

((a, b), A(a, b)) = inf_{g} Q_{g}(a, b).

Where Q_{g} is given by (0.9) and the infimum is taken over cylinder functions g. Proof. Define, on (X, π_{ρ_{β}, ρ_{ω}}), the functions

χ(v) = \frac{ρ_{ω}}{ρ} β(v) − \frac{ρ_{β}}{ρ} ω(v) and ζ(v) = β(v) + ω(v).

These are orthogonal, and in fact,

E[χ(u)f(η)] = 0  (3.3)

for any function f which does not depend on the colour at the site u, but only on the presence of a particle there, for such a function can be written

ζ(u)f_{1}(η) + (1 − ζ(u))f_{2}(η) where f_{1} and f_{2} are independent of η(u). Also we have for any f_{g} and u ∈ Z^{d},

E[(ζ(u + v) − ζ(u))τ^{u}f_{g}^{v}] = 0  (3.4)

for if the support of g contained in [−m, m]^{d} then D^{u,u+v} ∑_{x} τ^{x}g is a finite sum so E[(ζ(u + v) − ζ(u))D^{u,u+v} ∑ τ^{x}g]

= ∑ E[(ζ(u + v) − ζ(u))D^{u,u+v}τ^{x}g]
= ∑ E[(ζ(u + v) − ζ(u))τ^{x}g]
= E[∑(ζ(u + v − x) − ζ(u − x))g] = 0

because the variables within the support of g cancel. Our third remark is that

a(u, u + v)(ζ(u + v) − ζ(u)) = (ζ(u + v) − ζ(u))  (3.5)

Using (0.10), (0.11), (0.12) we can calculate

inf_{g} E[∑_{v} p(v)a(0, v)(v · a(β(v) − β(0)) + v · b(ω(v) − ω(0)) + f_{g}^{v})^{2}]

= inf_{g} E[∑_{v} p(v)a(0, v)(v · (a − b)(χ(v) − χ(0))

+ v · (\frac{ρ_{β}}{ρ} a + \frac{ρ_{ω}}{ρ} b)(ζ(v) − ζ(0)) + f_{g}^{v})^{2}]

= inf_{g} E[∑_{v} p(v)a(0, v)(v · (a − b)(χ(v) − χ(0)) + f_{g}^{v})^{2}]

+ 2 ∑_{v} p(v)(v · (\frac{ρ_{β}}{ρ} a + \frac{ρ_{ω}}{ρ} b))^{2}ρ(1 − ρ)

So it remains to show

inf_{g} E[∑_{v} p(v)a(0, v)(v · (a − b)(χ(v) − χ(0)) + f_{g}^{v})^{2}] = \frac{ρ_{β}ρ_{ω}}{ρ} ((a − b), D_{s}(a − b))

Since v · (a − b)(χ(v) − χ(0)) ∈ T , by the remarks prior to the Theorem the infimum over T ∩ E_{0} is the same as over E_{0}. So the above is equal to

inf_{ψ(ζ)} E[∑_{v} p(v)a(0, v)(v·(a − b)(χ(v)−χ(0)) + D^{0,v}(∑_{u} χ(u)τ^{u}ψ))^{2}]

which, by direct calculation,

= inf_{ψ(ζ)} E[∑_{v} p(v)a(0, v)(v · (a − b)(χ(v) − χ(0)) + χ(v)ψ(ζ^{0,v})

− χ(0)ψ(ζ) + χ(0)τ^{v}ψ(ζ^{0,v}) − χ(v)τ^{v}ψ(ζ) + ∑_{u≠0,v} χ(u)τ^{u}D^{u,u+v}ψ)^{2}].

By using (0.12) and then (0.10) this can be reduced to

inf_{ψ(ζ)} E[∑_{v} p(v)((1 − ζ(0))χ^{2}(v)(ψ(ζ^{0,v}) − τ^{v}ψ(ζ) + v · (a − b))^{2}

+ (1 − ζ(v))χ^{2}(0)(τ^{v}ψ(ζ^{0,v}) − ψ(ζ) − v · (a − b)))^{2}

+ ∑_{v} p(v) ∑_{u≠0,v} χ^{2}(u)(τ^{u}D^{u,u+v}ψ)^{2}].

We can take ψ(ζ) to be independent of ζ(0), so this is equal to

\frac{ρ_{β}ρ_{ω}}{ρ} inf_{ψ(ζ)} E[∑_{v} p(v)((1 − ζ(−v))(τ^{v}ψ − ψ + v · (a − b))^{2}

+ (1 − ζ(v))(τ^{−v}ψ − ψ − v · (a − b)))^{2} + ∑_{v} p(v) ∑_{u≠0,v}(τ^{u}D^{u,u+v}ψ)^{2}].

By Theorem 0.3, this is equal to \frac{ρ_{β}ρ_{ω}}{ρ} ((a − b), D_{s}(a − b)) , as desired. □

4. Compactness

Let ϕ(θ) and ψ(θ) be smooth test functions on the torus, T^{d}. We write a stochastic differential equation for \frac{1}{N^{d}} ∑_{u} ϕ(\frac{u}{N})β(u) + ψ(\frac{u}{N})ω(u):

d(\frac{1}{N^{d}} ∑_{u} ϕ(\frac{u}{N})β(u) + ψ(\frac{u}{N})ω(u))

= −\frac{1}{N^{d}} ∑_{u,v} a(u, u + v)((ϕ(\frac{u − v}{N}) − ϕ(\frac{u}{N}))(β(u + v) − β(u))

+ (ψ(\frac{u − v}{N}) − ψ(\frac{u}{N}))(ω(u + v) − ω(u)))(N^{2}p(v)dt + dM_{u,u+v}^{N^{2}p(v)})

= −\frac{1}{N^{d+1}} ∑_{u,v} a(u, u + v)(v · ∇ϕ(\frac{u}{N})(β(u + v) − β(u))

+ v · ∇ψ(\frac{u}{N})(ω(u + v) − ω(u)))(N^{2}p(v)dt + dM_{u,u+v}^{N^{2}p(v)}) + o(1)

summing by parts once, where M_{u,u+v}^{N^{2}p(v)}(t) = P_{u,u+v}^{N^{2}p(v)}(t) − N^{2}p(v)t, P_{u,u+v}^{N^{2}p(v)} independent Poisson processes with intensity N^{2}p(v). Notice that it is not possible to sum by parts a second time, i.e. the model is non–gradient. If we define the following vector valued measures on the torus,

γ_{β}^{N} = \frac{N}{N^{d}} ∑_{u,v} p(v)v ∫_{0}^{t} a(u, u + v)(β_{s}(u + v) − β_{s}(u))ds · δ\frac{u}{N}
γ_{ω}^{N} = \frac{N}{N^{d}} ∑_{u,v} p(v)v ∫_{0}^{t} a(u, u + v)(ω_{s}(u + v) − ω_{s}(u))ds · δ\frac{u}{N}
ν_{β}^{N} = \frac{1}{N^{d+1}} ∑_{u,v} v ∫_{0}^{t} a(u, u + v)(β_{s}(u + v) − β_{s}(u))dM_{u,u+v}^{N^{2}p(v)} · δ\frac{u}{N}
ν_{ω}^{N} = \frac{1}{N^{d+1}} ∑_{u,v} v ∫_{0}^{t} a(u, u + v)(ω_{s}(u + v) − ω_{s}(u))dM_{u,u+v}^{N^{2}p(v)} · δ\frac{u}{N} ,

then the stochastic differential equation can be written in integrated form

(< ϕ, μ_{β}^{N}(t) > + < ψ, μ_{ω}^{N}(t) >) − (< ϕ, μ_{β}^{N}(0) > + < ψ, μ_{ω}^{N}(0) >)

= − < (∇ϕ, γ_{β}^{N}(t)) > − < (∇ψ, γ_{ω}^{N}(t)) >
− < (∇ϕ, ν_{β}^{N}(t)) > − < (∇ψ, ν_{ω}^{N}(t)) > +o(1).  (4.1)

Let Q_{N} be the measure on D([0, T] → X_{N}) given by the non–equilibrium process, with initial distribution f_{N}^{0}π^{N}, and let P_{N} be the measure given by the equilibrium process, with initial distribution π^{N}. These are related by a simple convexity inequality,

E^{Q_{N}} [f] ≤ \frac{1}{N^{d}} log E^{P_{N}} [exp N^{d}f] + \frac{H(Q_{N}; P_{N})}{N^{d}},

called the entropy inequality. It is not hard to estimate

\frac{H(Q_{N}; P_{N})}{N^{d}} ≤ log 3.

Let M(T^{d}) denote the signed measures, and H_{n}(T^{d}) the Sobolev space of order n. Q_{N} and P_{N} induce measures Q̃_{N} and P̃_{N} on D([0, T] → M(T^{d})^{4d+2}), the set of (μ_{β}^{N}, μ_{ω}^{N}, γ_{β}^{N}, γ_{ω}^{N}, ν_{β}^{N}, ν_{ω}^{N}). If k > d/2 then M(T^{d}) ⊂ H_{−k}(T^{d}). If we prove tightness only in some H_{−k}(T^{d}) we do not really lose any generality. The μ_{β}^{N} and μ_{ω}^{N} are probability measures so if they converge to measures whose densities are the solutions of (0.1) weakly in H_{−k}(T^{d}), they also converge weakly in M(T^{d}). Before we start we need two estimates.

Lemma 1. For ν^{N} either ν_{β}^{N} or ν_{ω}^{N},

EQ̃_{N} [ sup_{0≤t≤T} < ϕ, ν^{N}(t) >^{2}] ≤ \frac{1}{N^{d}} CT \frac{1}{N^{d}} ∑_{u} ϕ^{2}(\frac{u}{N}).

Proof.

EQ̃_{N} [(\frac{1}{N^{d}} ∑_{u} ϕ(\frac{u}{N}) ∫_{0}^{T} a(u, u + v)(β(u + v) − β(u))dM_{u,u+v})^{2}]

= \frac{1}{N^{2d}} ∑_{u} ϕ^{2}(\frac{u}{N}) ∫_{0}^{T} EQ̃_{N} [a(u, u + v)(β(u + v) − β(u))^{2}]dt

≤ \frac{1}{N^{d}} CT \frac{1}{N^{d}} ∑_{u} ϕ^{2}(\frac{u}{N}) → 0 as N → ∞,

so EQ̃_{N} [< ϕ, ν^{N}(T) >^{2}] → 0. By Doob’s inequality, since < ϕ, ν^{N}(t) > is a martingale,

EQ̃_{N} [ sup_{0≤t≤T} < ϕ, ν^{N}(t) >^{2}] ≤ 4EQ̃_{N} [< ϕ, ν^{N}(T) >^{2}],

which proves the lemma for ν_{β}^{N}. The proof for ν_{ω}^{N} is analogous. □

Lemma 2. For γ^{N} either γ_{β}^{N} or γ_{ω}^{N},

EP̃_{N} [exp N^{d}α |<ϕ, γ^{N}(t)> − <ϕ, γ^{N}(s)>|] ≤ 2 exp | t−s | α^{2} ∑_{u} ϕ^{2}(\frac{u}{N})

Proof. By stationarity, if t > s,

EP̃_{N} [exp N^{d}α |< ϕ, γ^{N}(t) > − < ϕ, γ^{N}(s) >|]
= EP̃_{N} [exp N^{d}α |< ϕ, γ^{N}(t − s) >|]
≤ exp(t − s)Λ_{N}(ϕ)

by the spectral theorem and Feynman–Kac formula, where

Λ_{N}(ϕ) = sup_{densities f}\{Nα ∑_{u} ϕ(\frac{u}{N})E^{fπ^{N}} [a(u, u + v)(β(u + v) − β(u))]

− \frac{N^{2}}{2} D_{N}(\sqrt{f})\}

is the principal eigenvalue of L_{N} + \frac{N}{2} α ∑_{u} ϕ(\frac{u}{N})a(u, u + v)(β(u + v) − β(u)). Now, for the expectation with respect to π^{N},

E[a(u, u + v)(β(u + v) − β(u))f(η)]
= \frac{1}{2}E[a(u, u + v)(β(u + v) − β(u))(f(η^{u,u+v}) − f(η))]
= \frac{1}{2}E[a(u, u+v)(β(u+v)−β(u))(\sqrt{f}(η^{u,u+v})+\sqrt{f}(η))(\sqrt{f}(η^{u,u+v})−\sqrt{f}(η))]
≤ \frac{1}{2} \sqrt{E[(β(u+v)−β(u))^{2}2(f(η^{u,u+v})+f(η))]E[a(u, u+v)(D^{u,u+v}(\sqrt{f}))^{2}]}
≤ 2\sqrt{Eπ^{N} [a(u, u + v)(D^{u,u+v}(\sqrt{f}))^{2}]},

so

Λ_{N}(ϕ) ≤ sup_{densities f}\{∑_{u} αϕ(\frac{u}{N})N\sqrt{Eπ^{N} [a(u, u + v)(D^{u,u+v}(\sqrt{f}))^{2}]}

− \frac{N^{2}}{2} Eπ^{N} [a(u, u + v)(D^{u,u+v}(\sqrt{f}))^{2}\}

≤ Cα^{2} ∑_{u} ϕ^{2}(\frac{u}{N})

which proves the lemma for γ_{β}^{N}. The proof for γ_{ω}^{N} is analogous. □

Theorem 4.1. Q̃_{N} is tight as a sequence of measures on

D([0, T] → (H_{−(d+2)}(T^{d}))^{4d+2}).

Proof. It suffices to prove

I. lim_{l→∞} \overline{lim}_{N→∞} Q̃_{N}( sup_{0≤t≤T} ∥υ(t)∥_{H_{−(d+1)}} ≥ l) = 0

and, for every ϵ > 0,

II. lim_{δ→0} \overline{lim}_{N→∞} Q̃_{N}( sup_{|s−t|<δ^{0≤s<t≤T}} ∥υ(t) − υ(s)∥_{H_{−(d+2)}} > ϵ) = 0

for υ equals each of i. μ_{β}^{N}, ii. μ_{ω}^{N}, iii. {γ_{β}^{N}}_{i}, iv. {γ_{ω}^{N}}_{i}, v. {ν_{β}^{N}}_{i}, vi. {ν_{ω}^{N}}_{i} (see, for example, [S–V]). For i and ii

< ϕ, μ^{N} >≤ ∫_{T^{d}} ϕ(θ)dθ ≤ \sqrt{∫_{T^{d}} ϕ^{2}(θ)dθ} so I is obvious.

Next we prove I and II for v and vi. Consider the functions f_{n}(θ) = exp 2πin · θ for n = (n_{1}, …, n_{d}), n_{j} positive integers, which form an orthonormal basis of L^{2}(T^{d}). The functions

f_{n}^{k} = \frac{f_{n}}{(1 + n_{1}^{2} + ⋯ + n_{d}^{2})^{\frac{k}{2}}}

form an orthonormal basis of H_{−k}(T^{d}). Using Lemma 1 we get

EQ̃_{N} [ sup_{0≤t≤T} ∥ν^{N}(t)∥_{H_{−(d+1)}}^{2}] = EQ̃_{N} [∑_{n} sup_{0≤t≤T} < f_{n}^{d+1}ν^{N}(t) >^{2}]
≤ \frac{1}{N^{d}} CT ∑_{n} \frac{1}{N^{d}} ∑_{u}(f_{n}^{d+1}(\frac{u}{N}))^{2}
≤ \frac{1}{N^{d}} CT ∑_{n} \frac{1}{(1 + n_{1}^{2} + ⋯ + n_{d}^{2})^{d+1}} → 0.

I, II for v, vi follow from this by Tchebyshev’s inequality.

Now we prove I and II for iii and iv. To make use of Lemma 2 we need the lemma of Garsia, Rodemich, and Rumsey ([S–V] p.47) which states that for any continuous function f(t) if Ψ(x) is a strictly increasing function with Ψ(0) = 0 and lim_{x→∞} Ψ(x) = ∞ and

B = ∫_{0}^{T} ∫_{0}^{T} Ψ(\frac{| f(t) − f(s) |}{| t − s |^{1/2}})dsdt

then for 0 ≤ s < t ≤ T

| f(t) − f(s) |≤ 4 ∫_{0}^{(t−s)} Ψ^{−1}(\frac{4B}{u^{2}} )\frac{du}{\sqrt{u}}.

We will take

Ψ(x) = exp(N^{d}αx) − 1. By Lemma 2

EP̃_{N} [B] ≤ T^{2}(2 exp(α^{2} ∑_{u} ϕ^{2}(\frac{u}{N})) − 1).

For this Ψ it is not hard to show that for (t − s) ≤ 1

4 ∫_{0}^{(t−s)} Ψ^{−1}(\frac{4B}{u^{2}} )\frac{du}{\sqrt{u}} ≤ \frac{8\sqrt{t − s}}{N^{d}α} log(4B + (t − s)^{2}).

Then we can calculate

EP̃_{N} [exp \frac{N^{d}α}{8\sqrt{δ}} sup_{|t−s|<δ^{0≤s<t≤T}} |< ϕ, γ^{N}(t) > − < ϕ, γ^{N}(s) >|]

≤ EP̃_{N} [4B + δ^{2}]

≤ 8T^{2} exp(α^{2} ∑_{u} ϕ^{2}(\frac{u}{N}))

Integrating over α we get

EP̃_{N} [exp \frac{N^{d}}{Cδ} sup_{|t−s|<δ^{0≤s<t≤T}} \frac{|< ϕ, γ^{N}(t) > − < ϕ, γ^{N}(s) >|^{2}}{\frac{1}{N^{d}} ∑_{u} ϕ^{2}(\frac{u}{N})} ] ≤ 8T^{2}.

From this, using the entropy inequality, we can get

EQ̃_{N} [ sup_{|t−s|<δ^{0≤s<t≤T}} |< ϕ, γ^{N}(t) > − < ϕ, γ^{N}(s) >|^{2}] ≤ Cδ \frac{1}{N^{d}} ∑_{u} ϕ^{2}(\frac{u}{N}).

We can estimate the H_{−(d+1)} norm as before

EQ̃_{N} [ sup_{|t−s|<δ^{0≤s<t≤T}} ∥γ^{N}(t) − γ^{N}(s)∥_{H_{−(d+1)}}^{2}]

= EQ̃_{N} [∑_{n} sup_{|t−s|<δ^{0≤s<t≤T}} |< f_{n}^{d+1}, γ^{N}(t) > − < f_{n}^{d+1}, γ^{N}(s) >|^{2}]

≤ Cδ ∑_{n} \frac{1}{N^{d}} ∑_{u}(f_{n}^{d+1}(\frac{u}{N}))^{2}

≤ Cδ ∑_{n} \frac{1}{(1 + n_{1}^{2} + ⋯ + n_{d}^{2})^{d+1}} = C_{d}δ < ∞.

I, II for iii, iv follow by Tchebyshev’s inequality.

Finally to prove II for i and ii we use the stochastic differential equation and the result for iii, iv, v and vi. This completes the proof of compactness. □

5. Inequalities involving the Dirichlet form

Because of the exclusion, the operator L_{N} is degenerate, and we need some inequalities to keep things under control. These exploit the following simple fact: The exchange between sites x and y can be performed by a configuration independent sequence of excluded nearest neighbour exchanges, as long as two sites adjacent to x are unoccupied.

We start with some notation. For any l let B_{l}(u) be the box of side length 2l + 1

centred at u. Let ρβ^{l}(u) = \frac{1}{(2l+1)^{d}} ∑_{x∈B_{l}(u)} β(u) and ρω^{l}(u) = \frac{1}{(2l+1)^{d}} ∑_{x∈B_{l}(u)} ω(u) be

the densities of blue and white particles in that box. Also let ρ^{l} = ρβ^{l} + ρω^{l}. For the uncoloured process the macroscopic profile ρ evolves according to the heat equation.

We have assumed for the initial profiles ρ_{β}^{0}(θ) + ρ_{ω}^{0}(θ) = ρ^{0}(θ) < 1. Fix δ > 0 such that ρ^{0}(θ) < 1 − δ. Then at a later time we still have ρ(t, θ) < 1 − δ. Let A_{k} be the set of configurations where there are one or fewer unoccupied sites in B_{k}(0). It then follows from results in [K–O–V] that

\overline{lim}_{k→∞} \overline{lim}_{N→∞} E^{Q_{N}} [∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} τ^{u}1_{A_{k}}dt] = 0.  (5.1)

Lemma 3. For any function f on X_{N} define f̄ = ∑_{u} τ^{u}f. If z ∈ B_{l}(0) for some k < l < N then there is a universal constant C such that

Eπ^{N} [1_{X_{N} \A_{k}}(D^{0,z}(\sqrt{f̄}))^{2}] ≤ C \frac{k^{2d}(l + k^{d})^{2}}{N^{d}} D_{N}(\sqrt{f̄})

Proof. To each configuration we will associate a sequence of bonds, b_{1}, …, b_{n}, where b_{i} = u_{i}, u_{i} + v_{i}, with the following properties

i. η^{b_{1}···b_{n}} = η^{0,z}

ii. p(v_{i}) > 0

iii. a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}}) = 1

iv. No single bond is used more than 10 times.

Step I: For a configuration in X_{N} \A_{k} there must be at least two empty sites in B_{k}. Choose the closest unoccupied sites x_{1}, x_{2} to 0 in the norm | v |= max{v_{1}, …, v_{d}} (if there is more than one possibility choose the candidate closest in dimension 1, then dimension 2, etc.) Move x_{1} to (1, 0, …, 0) by first moving it in dimension d, then in dimension d − 1, etc. Next move x_{2} to (1, 1, 0, …, 0) in the same way, using bonds b_{m_{0}+1}, …, b_{m_{1}}.

Step II: Once two sites adjacent to 0 are unoccupied it is possible to choose a configuration independent sequence of bonds b_{m_{1}+1}, …, b_{m_{2}} such that η^{b_{m_{1}}···b_{m_{2}}} = η^{0,z}.

This sequence can be assumed to use bonds only in the “corridor” of width 2 around the line joining 0 to z by first joining (0, …, 0) to (0, …, 0, z_{d}), then (0, …, 0, z_{d}) to (0, …, z_{d−1}, z_{d}), etc. This corridor has maximum length dNϵ .

Step III: Move the empty sites back where they came from by reversing the order of exchanges in step I.

Of course the sequence of bonds is random, but all the bonds used lie inside B_{k} + the corridor, and the chain is less than 10(dl + k^{d}) bonds long. We have

E^{π}[(D^{0,z}\sqrt{f̄})^{2}1_{X_{N} \A_{k}}]

= E^{π}[(∑_{i=1}^{n} a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}})D^{u_{i},u_{i}+v_{i}} \sqrt{f̄}(η^{b_{1}···b_{i−1}}))^{2}]

≤ 10(dl + k^{d})E^{π}[∑_{i=1}^{n} a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}})(D^{u_{i},u_{i}+v_{i}} \sqrt{f̄}(η^{b_{1}···b_{i−1}}))^{2}]

= 10(dl+k^{d}) ∑_{y_{1},y_{2}∈B_{k}} E^{π}[1_{y_{2}=x_{2}^{y_{1}=x_{1}}} ∑_{i=1}^{n} a(u_{i}, u_{i}+v_{i})(η^{b_{1}···b_{i−1}})(D^{u_{i},u_{i}+v_{i}}\sqrt{f̄}(η^{b_{1}···b_{i−1}}))^{2}]

≤ 10(dl+k^{d}) ∑_{y_{1},y_{2}∈B_{k}} E^{π}[∑_{i=1}^{n} a(u_{i}, u_{i}+v_{i})(η^{b_{1}···b_{i−1}})(D^{u_{i},u_{i}+v_{i}}\sqrt{f̄}(η^{b_{1}···b_{i−1}}))^{2}]

= 10(dl+k^{d}) ∑_{y_{1},y_{2}∈B_{k}} E^{π}[∑_{i=1}^{n} a(u_{i}, u_{i}+v_{i})(η)(D^{u_{i},u_{i}+v_{i}}\sqrt{f̄}(η))^{2}]

≤ 10(dl + k^{d}) ∑_{y_{1},y_{2}∈[0,...,k]^{d}} E^{π}[  ∑_{y,y+v∈[0,...,k]^{d} + corridor} a(y, y + v)(D^{y,y+v}(\sqrt{f̄}))^{2}]

≤ 10(dl + k^{d})k^{2d}E^{π}[ ∑_{y,y+v∈[0,...,k]^{d} + corridor} a(y, y + v)(D^{y,y+v}(\sqrt{f̄}))^{2}]

≤ \frac{C10K^{2d}(dl + k^{d})^{2}}{N^{d}} D_{N}(\sqrt{f}),

which proves the lemma. □

Now we consider the action of L_{N} on the non-periodic hyperplane with N^{d}ρ_{β} blue particles and N^{d}ρ_{ω} white particles, with uniform measure u_{ρ_{β}, ρ_{ω}}^{N}. The Dirichlet form is

D_{N}(f) = Eu_{ρ_{β}, ρ_{ω}}^{N} [∑_{u,v} p(v)a(u, u + v)(D^{u,u+v}f)^{2}].

Lemma 4. For 1 ≤ p < 2 and ρ_{β} + ρ_{ω} < 1

∑_{u,v} Eu_{ρ_{β}, ρ_{ω}}^{N} [p(v) | D^{u,u+v}f |^{p}] ≤ C_{p}N^{d( \frac{2−p}{2} )}(D_{N}(f))^{\frac{p}{2}}

where the constant C_{p} depends only on ρ_{β} + ρ_{ω} and p.

Proof. One can make the exchange D^{u,u+v} by a configuration independent sequence of exchanges as long as two sites adjacent to u are unoccupied. So we find the closest two unoccupied sites x_{1} and x_{2} and move them by a sequence of exchanges to sites adjacent to u. We do this according to the same rule as in the previous lemma. The difference here is that we are no longer guaranteed unoccupied sites in a small box B_{k}(u). We have, with respect to the uniform measure u_{ρ_{β}, ρ_{ω}}^{N}

E[| D^{u,u+v}f |^{p}]

= ∑_{y_{1},y_{2}} E[1_{y_{2}=x_{2}^{y_{1}=x_{1}}} | ∑_{i=1}^{n} a(u_{i}, u_{i} + v_{i})(η^{b_{1},...,b_{i−1}})D^{u_{i},u_{i}+v_{i}}f(η^{b_{1},...,b_{i−1}}) |^{p}]

≤ ∑_{y_{1},y_{2}} n^{p−1} ∑_{i=1}^{n} E[1_{y_{2}=x_{2}^{y_{1}=x_{1}}}a(u_{i}, u_{i} + v_{i})(η^{b_{1},...,b_{i−1}}) | D^{u_{i},u_{i}+v_{i}}f(η^{b_{1},...,b_{i−1}}) |^{p}]

≤ ∑_{y_{1},y_{2}} n^{p−1} ∑_{i=1}^{n}(P [y_{1} = x_{1}, y_{2} = x_{2}])^{\frac{2−p}{2}}

× (E[a(u_{i}, u_{i} + v_{i})(η^{b_{1},...,b_{i−1}})(D^{u_{i},u_{i}+v_{i}}f(η^{b_{1},...,b_{i−1}}))^{2}])^{\frac{p}{2}} .

The path b_{1}, …, b_{n} is uniquely determined by y_{1}, y_{2} so, since u_{ρ_{β}, ρ_{ω}}^{N} is exchangeable, this is equal to

∑_{y_{1},y_{2}} n^{p−1}(P [y_{1} = x_{1}, y_{2} = x_{2}])^{\frac{2−p}{2}} ∑_{i=1}^{n}(E[a(u_{i}, u_{i} + v_{i})(η)(D^{u_{i},u_{i}+v_{i}}f(η))^{2}])^{\frac{p}{2}}

≤ ∑_{k_{1}≤k_{2}} ∑_{|y_{2}|=k_{2}^{|y_{1}|=k_{1}}}n^{p−1}(P [|x_{1} |=k_{1}, |x_{2} |=k_{2}])^{\frac{2−p}{2}} ∑_{i=1}^{n}(E[a(u_{i}, u_{i} + v_{i})(D^{u_{i},u_{i}+v_{i}}f)^{2}])^{\frac{p}{2}} .

It is not hard to show that P [| x_{1} |= k_{1}, | x_{2} |= k_{2}] is less than ρ^{− \frac{k^{d}_{2}}{2}} so

E[| D^{u,u+v}f |^{p}] ≤ ∑_{k_{1}≤k_{2}} ∑_{|y_{2}|=k_{2}^{|y_{1}|=k_{1}}} n^{p−1}ρ^{−( \frac{2−p}{4} )k_{2}^{d}} ∑_{i=1}^{n}(E[a(u_{i}, u_{i} + v_{i})(D^{u_{i},u_{i}+v_{i}}f)^{2}])^{\frac{p}{2}} .

Since each bond is used less than 10 times and the chain is less than Ck_{2} bonds long this is

≤ C ∑_{k_{1}≤k_{2}} ∑_{|y_{2}|=k_{2}^{|y_{1}|=k_{1}}} k_{2}^{p−1}ρ^{−( \frac{2−p}{4} )k_{2}^{d}} ∑_{y,y+v∈B_{k_{2}}(u)}(E[p(v)a(y, y + v)(D^{y,y+v}f)^{2}])^{\frac{p}{2}}

≤ C ∑_{k_{2}} k_{2}^{2d+p−2}ρ^{−( \frac{2−p}{4} )k_{2}^{d}} ∑_{y,y+v∈B_{k_{2}}(u)}(E[p(v)a(y, y + v)(D^{y,y+v}f)^{2}])^{\frac{p}{2}} .

Then

∑_{u,u+v} E[p(v) | D^{u,u+v}f |^{p}]

≤ C ∑_{u} ∑_{k_{2}} k_{2}^{2d+p−2}ρ^{−( \frac{2−p}{4} )k_{2}^{d}} ∑_{y,y+v∈B_{k_{2}}(u)}(E[p(v)a(y, y + v)(D^{y,y+v}f)^{2}])^{\frac{p}{2}}

≤ C ∑_{k_{2}} k_{2}^{3d+p−2}ρ^{−( \frac{2−p}{4} )k_{2}^{d}} ∑_{u,u+v}(E[p(v)a(u, u + v)(D^{u,u+v}f)^{2}])^{\frac{p}{2}}

≤ C ∑_{u,u+v}(E[p(v)a(u, u + v)(D^{u,u+v}f)^{2}])^{\frac{p}{2}}

≤ CN^{\frac{2−p}{2}} (D_{N}(f_{N}))^{\frac{p}{2}} . □

Now we estimate how much Dirichlet form is needed to perform a shift.

Lemma 5. Let l < N and G_{l} be a function which only depends on variables at sites in B_{l}(0). Then for ρ_{β} + ρ_{ω} < 1 there exists a constant C(ρ_{β} + ρ_{ω}) such that for all functions f

Eu_{ρ_{β}, ρ_{ω}}^{N} [f(τ^{e_{j}}(G_{l}1_{X_{N} \A_{l}}) − (G_{l}1_{X_{N} \A_{l}}))]

≤ Cl^{\frac{3d}{2}} ∥G_{l}∥_{2}\sqrt{∑_{y,y+v∈B_{l}(0)} Eu_{ρ_{β}, ρ_{ω}}^{N} [p(v)a(u, u + v)(D^{u,u+v}f)^{2}]},

and

Eu_{ρ_{β}, ρ_{ω}}^{N} [f^{2}(τ^{e_{j}}(G_{l}1_{X_{N} \A_{l}}) − (G_{l}1_{X_{N} \A_{l}}))]

≤ Cl^{\frac{3d}{2}} \sqrt{E[f^{2}(τ^{e_{j}}G_{l}^{2} + G_{l}^{2})]}\sqrt{∑_{y,y+v∈B_{l}(0)} Eu_{ρ_{β}, ρ_{ω}}^{N} [p(v)a(u, u + v)(D^{u,u+v}f)^{2}]}.

Proof. For the expectation with respect to u_{ρ_{β}, ρ_{ω}}^{N}

E[f(τ^{e_{j}}(G_{l}1_{X_{N} \A_{l}}) − (G_{l}1_{X_{N} \A_{l}}))] = E[(τ^{e_{j}}f − f)1_{X_{N} \A_{l}}G_{l}]
≤ ∥G_{l}∥_{2}\sqrt{E[(τ^{e_{j}}f − f)^{2}1_{X_{N} \A_{l}}]}

and

E[f^{2}(τ^{e_{j}}(G_{l}1_{X_{N} \A_{l}}) − (G_{l}1_{X_{N} \A_{l}}))]
= E[(τ^{e_{j}}f^{2} − f^{2})1_{X_{N} \A_{l}}G_{l}]
= E[(τ^{e_{j}}f − f)(τ^{e_{j}}f + f)1_{X_{N} \A_{l}}G_{l}]
≤ \sqrt{E[f^{2}(τ^{e_{j}}G_{l}^{2} + G_{l}^{2})]}\sqrt{E[(τ^{e_{j}}f − f)^{2}1_{X_{N} \A_{l}}]}

so the problem is to estimate

E[(τ^{e_{j}}f − f)^{2}1_{X_{N} \A_{l}}] ≤ Cl^{3d} ∑_{y,y+v∈B_{l}(0)} E[p(v)a(u, u + v)(D^{u,u+v}f)^{2}].

Once two sites adjacent to 0 are unoccupied we can find a configuration independent sequence of bonds of length less than Cl^{d} which performs the desired shift. Because we are in the set X_{N} \ A_{l} there are two unoccupied sites somewhere in B_{l}(0). Choose the two unoccupied sites x_{1} and x_{2} closest to 0 and move them adjacent to 0. Perform the desired shift, then move them back where they came from. Thus we have a sequence of bonds b_{1}, …, b_{n} with n ≤ Cl^{d} such that a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}}) = 1 and , for all f, f(η^{b_{1}···b_{n}}) = τ^{e_{j}}f(η). Now we can estimate

E[(τ^{e_{j}}f − f)^{2}1_{X_{N} \A_{l}}]

= ∑_{y_{1},y_{2}∈B_{l}(0)} E[1_{y_{2}=x_{2}^{y_{1}=x_{1}}}(∑_{i=1}^{n} a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}})D^{u_{i},u_{i}+v_{i}}f(η^{b_{1}···b_{i−1}}))^{2}]

≤ ∑_{y_{1},y_{2}∈B_{l}(0)} n ∑_{i=1}^{n} E[a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}})(D^{u_{i},u_{i}+v_{i}}f(η^{b_{1}···b_{i−1}}))^{2}]

= ∑_{y_{1},y_{2}∈B_{l}(0)} n ∑_{i=1}^{n} E[a(u_{i}, u_{i} + v_{i})(η)(D^{u_{i},u_{i}+v_{i}}f(η))^{2}]

≤ Cl^{3d} ∑_{y,y+v∈B_{l}(0)} E[p(v)a(u, u + v)(D^{u,u+v}f)^{2}]. □

If we are not guaranteed two empty sites in a small box B_{l} we can prove a slightly weaker version of the previous lemma.

Lemma 6. Let l < N and G_{l} be a function which only depends on variables at sites in B_{l}(0). For 1 ≤ p < 2 and ρ_{β} + ρ_{ω} < 1 there exists a constant C_{p}(ρ_{β} + ρ_{ω}) such that for all f

∑_{u} Eu_{ρ_{β}, ρ_{ω}}^{N} [f(τ^{u+e_{i}}G_{l} − τ^{u}G_{l})] ≤ C_{p}l^{d( \frac{p−1}{p})}N^{\frac{d}{2}} ∥G_{l}∥_{q}\sqrt{D_{N}(f)}

where \frac{1}{p} + \frac{1}{q} = 1.

Proof. Consider a fixed u.

E[f(τ^{u+e_{i}}G_{l} − τ^{u}G_{l})] = E[(τ^{u+e_{i}}f − τ^{u}f)G_{l}]
≤ ∥G_{l}∥_{q}(E[| τ^{u+e_{i}}f − τ^{u}f |^{p}])^{\frac{1}{p}} .

We can find the closest two unoccupied sites to u, x_{1} and x_{2}, and move them adjacent to u as in the previous lemma. The difference is that we may have to look far for them. We get

E[| τ^{u+e_{i}}f − τ^{u}f |^{p}]

= ∑_{y_{1},y_{2}∈B_{l}(0)} E[1_{y_{2}=x_{2}^{y_{1}=x_{1}}} | ∑_{i=1}^{n} a(u_{i}, u_{i} + v_{i})(η^{b_{1}···b_{i−1}})D^{u_{i},u_{i}+v_{i}}f(η^{b_{1}···b_{i−1}}) |^{p}].

We can estimate this as in Lemma 4 to be

≤ ∑_{k_{1}≤k_{2}} ∑_{|y_{2}|=k_{2}^{|y_{1}|=k_{1}}} n^{p−1}ρ^{−( \frac{2−p}{4} )(k_{2})^{d}} ∑_{i=1}^{n}(E[a(u_{i}, u_{i} + v_{i})(D^{u_{i},u_{i}+v_{i}}f)^{2}])^{\frac{p}{2}} .

The chain is less than C(l^{d} + k_{2}) bonds long so this is

≤ C ∑_{k_{1}≤k_{2}} ∑_{|y_{2}|=k_{2}^{|y_{1}|=k_{1}}}(l^{d} + k_{2})^{p−1}ρ^{−( \frac{2−p}{4} )(k_{2})^{d}} ∑_{y,y+v∈B_{k_{2}}(u)}(E[a(y, y + v)(D^{y,y+v}f)^{2}])^{\frac{p}{2}}

≤ C ∑_{k_{2}}(k_{2})^{2d−1}(l^{d} + k_{2})^{p−1}ρ^{−( \frac{2−p}{4} )(k_{2})^{d}} ∑_{y,y+v∈B_{k_{2}}(u)}(E[a(y, y + v)(D^{y,y+v}f)^{2}])^{\frac{p}{2}} .

So we have

∑_{u} E[f(τ^{u+e_{i}}G_{l} − τ^{u}G_{l})]

≤ C∥G_{l}∥_{q} ∑_{u}(∑_{k_{2}}(k_{2})^{2d−1}(l^{d}+k_{2})^{p−1}ρ^{−( \frac{2−p}{4} )(k_{2})^{d}} ∑_{y,y+v∈B_{k_{2}}(u)} (E[a(y, y+v)(D^{y,y+v}f)^{2}])^{\frac{p}{2}} )^{\frac{1}{p}}

≤ C∥G_{l}∥_{q} ∑_{k_{2}}(k_{2})^{\frac{2d−1}{p}} (l^{d} + k_{2})^{\frac{p−1}{p}} ρ^{−( \frac{2−p}{4p})(k_{2})^{d}} ∑_{y,y+v}(E[a(y, y + v)(D^{y,y+v}f)^{2}])^{\frac{1}{2}}

≤ C_{p}l^{d( \frac{p−1}{p})}N^{\frac{d}{2}} ∥G_{l}∥_{q}\sqrt{D_{N}(f)} □

6. Local equilibrium distributions

In this section we show that the microscopic densities of blue and white particles do not fluctuate on a macroscopic scale. The particular result we need is that for any continuous function Q on T^{d} × [0, 1] × [0, 1],

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{λ_{N}}{N^{d}} = 0

where

λ_{N} = sup_{densitiesf_{N}}\{E^{f_{N}π^{N}} [∑_{u}(Q(\frac{u}{N} , ρ_{β}^{k}(u), ρ_{ω}^{k}(u)) − Q(\frac{u}{N} , ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))

− mτ^{u}1_{A_{k}})] − N^{2}D_{N}(\sqrt{f_{N}})\}.

An average over a block of side length Nϵ is just an average of averages over small blocks of side length k, and Q is uniformly continuous. Furthermore the result is known for functions only depending on ρ = ρ_{β} + ρ_{ω}. Therefore it is enough to prove the following

Lemma 7.

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} sup_{z∈B_{Nϵ}(0)} \frac{λ̃_{N}}{N^{d}} = 0

where

λ̃_{N} = sup_{densitiesf_{N}}\{E^{f_{N}π^{N}} [∑_{u}(| ρ_{β}^{k}(u + z) − ρ_{β}^{k}(u) |) − mτ^{u}1_{A_{k}})] − N^{2}D_{N}(\sqrt{f_{N}})\}

Proof. Since the first term in the supremum is O(N^{d}) the lemma is equivalent to

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} sup_{E^{f_{N}π^{N}} [m ∑_{u} τ^{u}1_{A_{k}}]≤C_{2}N^{d}^{D_{N} (\sqrt{f_{N}})≤C_{1}N^{d−2}}} sup_{z∈B_{Nϵ}(0)}

E^{f_{N}π^{N}} [\frac{1}{N^{d}} ∑_{u} | ρ_{β}^{k}(u + z) − ρ_{β}^{k}(u) |] = 0.

Define

B_{C_{1},C_{2}}^{N} = \{f_{N} : D_{N}(\sqrt{f_{N}}) ≤ C_{1}N^{d−2}, E^{f_{N}π^{N}} [m ∑_{u} τ^{u}1_{A_{k}}] ≤ C_{2}N^{d}\}.

We can project the density f_{N} onto B_{k}(u + z) × B_{k}(u) and average the result over all u. Write f̄ = \frac{1}{N^{d}} ∑_{u} τ^{u}f_{N}^{k} and

B̄_{C_{1},C_{2}}^{N,k} = {f̄: f_{N} ∈ B_{C_{1},C_{2}}^{N}}.

Then we need

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} sup_{f∈B̄_{C_{1},C_{2}}^{N,k}} E^{fπ}[| ρ_{β}^{k}(z) − ρ_{β}^{k}(0) |] = 0

where π = π^{(2k+1)^{d}} ×π^{(2k+1)^{d}} is the projection of π^{N}. If f ∈ B̄_{C_{1},C_{2}}^{N,k} we have E^{fπ}[1_{A_{k}}] ≤ \frac{C_{1}}{m} → 0. Since the integrand is less than 2 we have

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} sup_{f∈B̄_{C_{1},C_{2}}^{N,k}} E^{fπ}[1_{A_{k}} | ρ_{β}^{k}(z) − ρ_{β}^{k}(0) |] = 0.

Then we need

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} sup_{f∈B̄_{C_{1},C_{2}}^{N,k}} E^{fπ}[1_{X_{k}\A_{k}} | ρ_{β}^{k}(z) − ρ_{β}^{k}(0) |] = 0.

Let A_{k,ϵ} be the limit points of f̄ as N, | z |→ ∞ with | z |≤ Nϵ. We need to show

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} sup_{f∈A_{k,ϵ}} E^{fπ}[1_{X_{k}\A_{k}} | ρ_{β}^{k}(z) − ρ_{β}^{k}(0) |] = 0.

So we need to study the set A_{k,ϵ}. Define the following functionals:

I_{k}^{0}(f) = E^{π}[ ∑_{u,u+v∈B_{k}(0)} p(v)a(u, u + v)(D^{u,u+v}(\sqrt{f}))^{2}]

I_{k}^{z}(f) = I_{k}^{0}(τ^{z}f)

I_{k}^{0,z}(f) = E^{π}[(D^{0,z}(\sqrt{f}))^{2}].

If f_{N} is a density with D_{N}(\sqrt{f_{N}}) ≤ C_{1}N^{d−2} then, by convexity,

I_{k}^{0}(f̄) = I_{k}^{z}(f̄) ≤ \frac{k^{d}}{N^{d}} D_{N}(\sqrt{f_{N}}) ≤ C_{1} \frac{k^{d}}{N^{2}} .

By Lemma 3

I_{k}^{0,z}(1_{X_{k}\A_{k}} f̄) ≤ C_{3} \frac{k^{2d}(Nϵ + k^{d})^{2}}{N^{d}} D_{N}(\sqrt{f_{N}}) ≤ C_{3} \frac{k^{2d}(Nϵ + k^{d})^{2}}{N^{2}} .

For f ∈ A_{k,ϵ},

I_{k}^{z}(f) = 0
I_{k}^{0,z}(1_{X_{k}\A_{k}}f) ≤ C_{3}k^{2d}ϵ^{2}

If we let A_{k} be the limit points of A_{k,ϵ} as ϵ → 0 then for f ∈ A_{k} we have I_{k}^{0}(f) = I_{k}^{z}(f) = I_{k}^{0,z}(1_{X_{k}\A_{k}}f) = 0 and we need

\overline{lim}_{m→∞} \overline{lim}_{k→∞} sup_{f∈A_{k}} E^{fπ}[1_{X_{k}\A_{k}} | ρ_{β}^{k}(z) − ρ_{β}^{k}(0) |] = 0.

Clearly the supremum is achieved over f concentrated on X_{k} \ A_{k}. For such an f we have I_{k}^{0}(f) = I_{k}^{z}(f) = I_{k}^{0,z}(f) = 0 which means that the density f is unchanged by exchanging any two variables η(u) and η(u + v) with u, u + v ∈ B_{k}(0) or B_{k}(z) and p(v) > 0, or by exchanging η(0) and η(z). From this we deduce that the density is unchanged by any permutation of the variables {η(x)}_{x∈B_{k}(0)×B_{k}(z)}. Therefore the measures fπ are mixtures of trinomial distributions, so we can estimate the above by

\overline{lim}_{k→∞} sup_{0≤i≤2(2k+1)^{d}} E^{π_{\frac{i}{2(2k+1)^{d}}}} [| ρ_{β}^{k}(z) − ρ_{β}^{k}(0) |],

where π_{\frac{i}{2(2k+1)^{d}}} is a trinomial with P [β(u)] = \frac{i}{2(2k+1)^{d}}. This can easily be estimated by Schwarz’s inequality to be

≤ \overline{lim}_{k→∞} sup_{0≤i≤2(2k+1)^{d}} \frac{1}{(2k + 1)^{d}} (\frac{i}{2(2k + 1)^{d}} (1 − \frac{i}{2(2k + 1)^{d}})) = 0. □

7. Second order perturbations

We start this section with a general fact about finite state Markov processes. Let L be the generator of an ergodic Markov process on a finite state space, reversible with respect to υ, and V a potential with E^{υ}[V] = 0. Let λ_{N} be the principal eigenvalue of L + \frac{1}{N} V, and define

σ_{V}^{2} = lim_{t→∞} E^{υ}[(\frac{1}{\sqrt{t}} ∫_{0}^{t} V (η_{s})ds)^{2}].

It is fairly straightforward to show that

\overline{lim}_{N→∞} N^{2}λ_{N} ≤ \frac{σ_{V}^{2}}{2} .  (7.1)

This fact provides the basis for the next theorem. Let V (θ, s, {η(x)}_{x∈B_{l}(0)}) be a function on T^{d}×[0, T]×{−1, 0, 1}^{B_{l}}, smooth in θ, with the property that the principal eigenvalue of

L_{N} + N ∑_{u} V (θ, s, {η(x)}_{x∈B_{l}(u)})

is bounded by BN^{d}, and such that V = 0 if all sites in B_{l}(u) are occupied. Let Q(θ, s, ρ_{β}, ρ_{ω}) be a continuous function on T^{d} × [0, T] × [0, 1] × [0, 1]. Define

σ_{k}^{2}(ρ_{β}, ρ_{ω}) = \frac{1}{(2(k−l)+1)^{d}} lim_{t→∞} E[(\frac{1}{\sqrt{t}} ∫_{0}^{t} ∑_{v∈B_{k−l}(0)}V (θ_{0}, s_{0}, \{η_{s}(x)\}_{x∈B_{l}(v)})ds)^{2}]

for ρ_{β}, ρ_{ω} taking the discrete values \frac{n}{(2k+1)^{d}} , \frac{m}{(2k+1)^{d}} with ρ_{β} + ρ_{ω} < 1. Here η_{s} is the non-speeded up process, i.e. with generator L̂_{2k+1} =^{def} \frac{1}{(2k+1)^{2}} L_{2k+1} on the lattice of side length 2k + 1 starting with uniform measure u_{ρ_{β}, ρ_{ω}}^{k}.

Theorem 7.1. If Q(ρ_{β}, ρ_{ω}) ≥ \overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}} σ_{k}^{2}(ρ_{β}^{k}, ρ_{ω}^{k}) , then

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{1}{N^{d}} log E^{P_{N}} [exp ∫_{0}^{T} F_{N,ϵ,k,m}(s)ds] ≤ 0

where

F_{N,ϵ,k,m}(s) = ∑_{u} NV (\frac{u}{N} , s, \{η(x)\}_{x∈B_{l}(u)}) − \frac{1}{2}Q(\frac{u}{N} , s, ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u)) − τ^{u}m1_{A_{k}}

Proof. Since P_{N} is the stationary process with reversible initial distribution π^{N} an application of the Feynman-Kac formula and the spectral theorem allows us to estimate

\frac{1}{N^{d}} log E^{P_{N}} [exp ∫_{0}^{T} F_{N,ϵ,k,m}(s)ds] ≤ ∫_{0}^{T} \frac{Λ_{N,ϵ,k,m}(s)}{N^{d}}ds

where Λ_{N,ϵ,k,m}(s) is the principal eigenvalue of L_{N} + F_{N,ϵ,k,m}(s) . The assumption on V implies that \frac{Λ_{N,ϵ,k,m}(s)}{N^{d}} ≤ B so by the dominated convergence theorem all we need is

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{Λ_{N,ϵ,k,m}(s)}{N^{d}} ≤ 0

for each fixed s. Λ_{N,ϵ,k,m}(s) is given by a variational formula

Λ_{N,ϵ,k,m}(s) = sup_{f_{N}}\{E^{f_{N}π^{N}} [F_{N,ϵ,k,m}(s)] − N^{2}D_{N}(\sqrt{f_{N}})\}.

Dropping the s we can write Λ_{N,ϵ,k,m} ≤ Λ_{N,k,1−γ}^{1} + Λ_{N,ϵ,k,m,γ}^{2} where

Λ_{N,k,1−γ}^{1} = sup_{f_{N}}\{E^{f_{N}π^{N}} [∑_{u} NV (\frac{u}{N} , \{η(x)\}_{x∈B_{l}(u)}) − \frac{1}{2}Q(\frac{u}{N} , ρ_{β}^{k}(u), ρ_{ω}^{k}(u))]

− N^{2}(1 − γ)D_{N}(\sqrt{f_{N}})\},

Λ_{N,ϵ,k,m,γ}^{2} = sup_{f_{N}}\{E^{f_{N}π^{N}} [∑_{u} Q(\frac{u}{N} , ρ_{β}^{k}(u), ρ_{ω}^{k}(u)) − Q(\frac{u}{N} , ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))

− τ^{u}m1_{A_{k}}] − N^{2}γD_{N}(\sqrt{f_{N}})\}.

We have shown in Lemma 7 that for any γ > 0

\overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{Λ_{N,ϵ,k,m,γ}^{2}}{N^{d}} = 0.

Next we write Λ_{N,k,1−γ}^{1} ≤ Λ_{N,k,1−γ}^{3} + Λ_{N,k}^{4} where

Λ_{N,k,1−γ}^{3} = sup_{f_{N}}\{E^{f_{N}π^{N}} [∑_{u} \frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} NV (\frac{u}{N} , \{η(x)\}_{x∈B_{l}(u+v)})

− \frac{1}{2}Q(\frac{u}{N} , ρ_{β}^{k}(u), ρ_{ω}^{k}(u))] − N^{2}(1 − γ)D_{N}(\sqrt{f_{N}})\},

Λ_{N,k}^{4} = sup_{f_{N}}\{E^{f_{N}π^{N}} [N ∑_{u} \frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} V (\frac{u − v}{N} , \{η(x)\}_{x∈B_{l}(u)})

− V (\frac{u}{N} , \{η(x)\}_{x∈B_{l}(u)})]\}.

If we use a two term Taylor expansion of V,

\frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} V (\frac{u − v}{N}) − V (\frac{u}{N})

= \frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} \frac{v}{N} \frac{∂V}{∂θ} (\frac{u}{N}) + \frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{l}(0)}(\frac{v}{N})^{2} \frac{1}{2} \frac{∂^{2}V}{∂θ^{2}} (θ̃),

the first order terms cancel by symmetry, so we can estimate

Λ_{N,k}^{4} ≤ \frac{(k − l)^{3}}{(k − l)^{d}} \frac{1}{N^{2}} N∥\frac{∂^{2}V}{∂θ^{2}} ∥_{∞}N^{d} = O(N^{d−1}).

It remains to show that

\overline{lim}_{γ→0} \overline{lim}_{k→∞} \overline{lim}_{N→∞} \frac{Λ_{N,k,1−γ}^{3}}{N^{d}} ≤ 0.

Now

\frac{Λ_{N,k,1−γ}^{3}}{N^{d}} ≤ \frac{1}{N^{d}} ∑_{u} sup_{f_{N}}\{E^{f_{N}π^{N}} [\frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} NV (\frac{u}{N} , \{η(x)\}_{x∈B_{l}(u+v)})

− \frac{1}{2}Q(\frac{u}{N} , ρ_{β}^{k}(u), ρ_{ω}^{k}(u))] − N^{2}(1 − γ)D_{N}(\sqrt{f_{N}})\}.

If we project f_{N} onto B_{k}(u) this becomes

\frac{1}{N^{d}} ∑_{u} sup_{f}\{E^{fπ^{k}}[\frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} NV (\frac{u}{N} , \{η(x)\}_{x∈B_{l}(u+v)})

− \frac{1}{2}Q(\frac{u}{N} , ρ_{β}^{k}(u), ρ_{ω}^{k}(u))] − \frac{N^{2}(1 − γ)}{(2k + 1)^{d}}D_{k}(\sqrt{f})\}.

Now we project onto the “hyperplane” of ρ_{β}(2k + 1)^{d} blue and ρ_{ω}(2k + 1)^{d} white particles, with uniform measure u_{ρ_{β}, ρ_{ω}}^{k}. Since there are only a finite number of such hyperplanes for each fixed k, the above is less than or equal to

\frac{1}{N^{d}} ∑_{u} sup_{ρ_{β},ρ_{ω}} sup_{f}\{E^{fu_{ρ_{β}, ρ_{ω}}^{k}} [\frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} NV (\frac{u}{N} , \{η(x)\}_{x∈B_{l}(u+v)})

− \frac{1}{2}Q(\frac{u}{N} , ρ_{β}, ρ_{ω})] − \frac{N^{2}(1 − γ)}{(2k + 1)^{d}}D_{k}(\sqrt{f})\}.

≤ sup_{θ} sup_{ρ_{β},ρ_{ω}} sup_{f}\{E^{fu_{ρ_{β}, ρ_{ω}}^{k}} [\frac{1}{(2(k−l)+1)^{d}} ∑_{v∈B_{k−l}(0)} NV (θ, \{η(x)\}_{x∈B_{l}(v)})

− \frac{1}{2}Q(θ, ρ_{β}, ρ_{ω})] − \frac{N^{2}(1 − γ)}{(2k + 1)^{d}}D_{k}(\sqrt{f})\}.

≤ sup_{θ} sup_{ρ_{β},ρ_{ω}}\{\frac{N^{2}(1 − γ)}{(2k + 1)^{d}}λ_{N} − \frac{1}{2}Q(θ, ρ_{β}, ρ_{ω})\}

where λ_{N} is the principal eigenvalue of

L̂_{2k+1} + \frac{1}{N} \frac{(2k + 1)^{d}}{(2(k − l) + 1)^{d}(1 − γ)} ∑_{v∈B_{k−l}(0)} V (θ, \{η(x)\}_{x∈B_{l}(v)}).

By (0.15) ,

\overline{lim_{γ→0}} \overline{lim}_{k→∞} \overline{lim}_{N→∞} \frac{Λ_{N,k,1−γ}^{3}}{N^{d}} ≤ 0.

which proves the theorem. □

8. Spectral gap

The estimates in this section concern functions of configurations on the non-periodic lattice of N^{d} sites. For the first estimate, consider the colourless configuration space {0, 1}^{T_{N}} with uniform measure u_{ρ}^{N}. The operator

L_{N}^{1}(f) = \frac{1}{N^{d}} ∑_{x,y} D^{x,y}f =^{def} \frac{1}{N^{d}} ∑_{x,y}(f(ζ^{x,y}) − f(ζ))

is the generator of the process where exchanges occur between any two sites.

Lemma 8. For any function f of mean 0,

Eu_{ρ}^{N} [f^{2}] ≤ \frac{1}{N^{d}} Eu_{ρ}^{N} [∑_{x,y}(D^{x,y}f)^{2}].

Proof. L^{2}(u_{ρ}^{N}) has dimension ( \begin{matrix}N^{d} \\ ρN^{d} \end{matrix} ). For integers n ≤ ρN^{d}, a polynomial of degree n is a function of the form ∑_{j=0}^{n} a_{j} ∏_{i=1}^{j} ζ(u_{ij}). Let U_{n} be the subspace generated by polynomials of degree ≤ n. If V_{n} is the orthogonal complement of U_{n−1} in U_{n} then

L^{2} = V_{0} ⊕ ⋯ ⊕ V_{ρN^{d}},

and each V_{n} has dimension ( \begin{matrix}N^{d} \\ n \end{matrix} ) − ( \begin{matrix}N^{d} \\ n − 1 \end{matrix} ). Applying L_{N}^{1} to ∏_{i=1}^{n} ζ(u_{i})

L_{N}^{1}(∏_{i=1}^{n} ζ(u_{i})) = −n\frac{N^{d} − (n − 1)}{N^{d}}(∏_{i=1}^{n} ζ(u_{i})) + ∑_{j=1}^{n}(ρ − \frac{(n − 1)}{N^{d}}) ∏_{i≠j^{i=1}}^{n} ζ(u_{i}),

so that if f ∈ V_{n}

L_{N}^{1}(f) = −n\frac{N^{d} − (n − 1)}{N^{d}}f + g

where g ∈ U_{n−1}. We can find h ∈ U_{n−1} such that

(L_{N}^{1} + n\frac{N^{d} − (n − 1)}{N^{d}})h = −g.

Then f + h is an eigenvector with eigenvalue −n\frac{N^{d}−(n−1)}{N^{d}}. Let W_{n} = \{f + h, f ∈ V_{n}}. Since h ∈ U_{n−1} the W_{n} still span L^{2}, and since L_{N}^{1} is self–adjoint the W_{n} are orthogonal. Thus W_{n} are the eigenspaces, and the eigenvalues are 0, −1, −2 + \frac{1}{N^{d}} , …, −ρN^{d} + \frac{ρN^{d}−1}{N^{d}}. □

Next we consider our simple exclusion process, restricted to {0, 1}^{T_{N}}.

Lemma 9. There exists a constant C such that for 0 ≤ ρ ≤ 1, and any function f of mean 0,

Eu_{ρ}^{N} [f^{2}] ≤ CN^{2}Eu_{ρ}^{N} [ ∑_{u,u+v} p(v)(D^{u,u+v}f)^{2}].

Proof. Connect sites x and y by a chain of bonds b_{i} = u_{i}, u_{i} + v_{i}; i = 1, …, n so that ζ^{x,y} = ζ^{b_{1}···b_{n}}, in the manner described in Lemma 4. We move x to y first along dimension 1, then dimension 2, etc. It is not hard to see that any fixed bond is used for at most d(\frac{N}{2} )^{d+1} pairs x, y. Then

\frac{1}{N^{d}} ∑_{x,y} E[(D^{x,y}f(ζ))^{2}] ≤ \frac{1}{N^{d}} ∑_{x,y} E[(∑_{i=1}^{n} D^{u_{i},u_{i}+v_{i}}f(ζ^{b_{1}···b_{i−1}}))^{2}]
≤ \frac{1}{N^{d}} ∑_{x,y} dN ∑_{i=1}^{n} E[(D^{u_{i},u_{i}+v_{i}}f(ζ^{b_{1}···b_{i−1}}))^{2}]
= \frac{1}{N^{d}} ∑_{x,y} dN ∑_{i=1}^{n} E[(D^{u_{i},u_{i}+v_{i}}f(ζ))^{2}]
≤ CN^{2} ∑_{u,u+v} E[p(v)(D^{u,u+v}f(ζ))^{2}].

Using the previous lemma we are done. □

Lemma 10. There exists a constant C such that for any 0 ≤ ρ ≤ 1, y ∈ T_{N}, and f with Eu_{ρ}^{N} [f | ζ(y) = 1] = 0,

Eu_{ρ}^{N} [f^{2} | ζ(y) = 1] ≤ CN^{2}Eu_{ρ}^{N} [ ∑_{u,u+v≠x,y} p(v)(D^{u,u+v}f)^{2} | ζ(y) = 1].

Proof. We do the same thing as in the previous lemma except that whenever a bond containing y comes up we go around it. For example, replace (y − e_{i}, y), (y, y + e_{i}) by (y−e_{i}, y−e_{i}−e_{i+1}), (y−e_{i}−e_{i+1}, y−e_{i+1}), (y−e_{i+1}, y−e_{i+1}+e_{i}), (y−e_{i+1}+e_{i}, y+e_{i}). The only effect is to increase the size of the constant C by a fixed amount. □

Now we come to the main result of this section. The generator L_{N} of the coloured simple exclusion process preserves the linear subspace T ⊂ L^{2}(u_{ρ_{β}, ρ_{ω}}^{N}) consisting of functions of the form f = ∑_{u}(aβ(u) + bω(u))F_{u}(ζ). We now estimate the spectral gap of L_{N} as an operator on T .

Lemma 11. For 0 < ρ_{β} + ρ_{ω} < 1 there exists a constant C(ρ_{β} + ρ_{ω}) such that for any function of the form f = ∑_{u}(aβ(u) + bω(u))F_{u}(ζ) with mean 0,

Eu_{ρ_{β}, ρ_{ω}}^{N} [f^{2}] ≤ CN^{2}Eu_{ρ_{β}, ρ_{ω}}^{N} [ ∑_{u,u+v} p(v)a(u, u + v)(D^{u,u+v}f)^{2}].

Proof. Rewrite ∑_{u}(aβ(u) + bω(u))F_{u} as

∑_{u}((a − b)χ(u) + (\frac{ρ_{β}}{ρ} a + \frac{ρ_{ω}}{ρ} b)ζ(u))F_{u} = f_{χ} + f_{ζ}.

Then for the expectation with respect to u_{ρ_{β}, ρ_{ω}}^{N},

E[(f_{χ} + f_{ζ})^{2}] = E[f_{χ}^{2}] + E[f_{ζ}^{2}]

and

E[ ∑_{u,u+v} p(v)a(u, u + v)(D^{u,u+v}(f_{χ} + f_{ζ}))^{2}]

= E[ ∑_{u,u+v} p(v)a(u, u + v)(D^{u,u+v}f_{χ})^{2}] + E[ ∑_{u,u+v} p(v)(D^{u,u+v}f_{ζ})^{2}],

so by Lemma 9 we only need to prove the estimate for f of the form f = ∑_{u} χ(u)F_{u}(ζ). Replacing F_{u} by F_{u} − F for any function F does not affect such an f, so we can take F_{u}(ζ) = 0 if ζ(u) = 0, and ∑_{u} F_{u}(ζ) = 0. Calculation yields

E[(∑_{y} χ(y)F_{y})^{2}] = (\frac{ρ_{β}ρ_{ω}}{ρ^{2}} + \frac{ρ_{β}ρ_{ω}}{ρ(ρN^{d} − 1)}) ∑_{y} E[F_{y}^{2}]

and

E[ ∑_{u,u+v} p(v)a(u, u + v)(D^{u,u+v}(∑_{y} χ(y)F_{y}))^{2}]

= (\frac{ρ_{β}ρ_{ω}}{ρ^{2}} + \frac{ρ_{β}ρ_{ω}}{ρ(ρN^{d} − 1)})E[ ∑_{u,u+v} p(v)( ∑_{y≠u,u+v}(D^{u,u+v}F_{y})^{2}

+ 2(1 − ζ(u + v))(F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ))^{2})].

Here we have used repeatedly that ∑_{u} F_{u} = 0 and that for any colourblind function

g(ζ), E[χ^{2}(u)g] = \frac{ρ_{β}ρ_{ω}}{ρ^{2}} E[ζ(u)g], and, for u ≠ v, E[χ(u)χ(v)g] = −\frac{ρ_{β}ρ_{ω}}{ρ(ρN^{d}−1)}E[ζ(u)ζ(v)g].

The problem is now reduced to showing

∑_{y} E[F_{y}^{2}] ≤ CN^{2}E[ ∑_{u,u+v} p(v)( ∑_{y≠u,u+v}(D^{u,u+v}F_{y})^{2}

+ 2(1 − ζ(u + v))(F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ))^{2})].

By Lemma 10 we have

E[F_{y}^{2}] − (E[F_{y}])^{2} ≤ CN^{2}E[ ∑_{u,u+v≠y} p(v)(D^{u,u+v}F_{y})^{2}]

so the problem is to estimate

∑_{y}(E[F_{y}])^{2} = \frac{1}{2N^{d}} ∑_{x,y}(E[F_{x} − F_{y}])^{2}

≤ CN^{2}E[ ∑_{u,u+v} p(v)( ∑_{y≠u,u+v}(D^{u,u+v}F_{y})^{2}+2(1−ζ(u+v))(F_{u+v}(ζ^{u,u+v})−F_{u}(ζ))^{2})].

It is easy to see that

\frac{1}{2N^{d}} ∑_{x,y}(E[F_{x} − F_{y}])^{2} ≤ CN^{2} ∑_{u,u+v}(E[p(v)(F_{u+v} − F_{u})])^{2},

where C = max_{v∈S}\{\frac{1}{p(v)}\}. Fix u, u + v. To each configuration we will associate a sequence of bonds b_{1}, …, b_{n} such that ζ^{b_{1}···b_{n}}(u + v) = 0. Pick the unoccupied site x_{u} nearest to u + v and choose b_{1}, …, b_{n} as in Step I of Lemma 3. No b_{i} need have u or u + v as an endpoint except b_{n}. Now

E[F_{u+v} − F_{u}] = E[F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ)]

= E[∑_{i=1}^{n}(D^{b_{i}}F_{u}(ζ^{b_{1}···b_{i−1}}) − D^{b_{i}}F_{u+v}((ζ^{u,u+v})^{b_{1}···b_{i−1}}))

+ (1 − ζ^{b_{1}···b_{n}}(u + v))(F_{u+v}((ζ^{u,u+v})^{b_{1}···b_{n}}) − F_{u}(ζ^{b_{1}···b_{n}}))]

= ∑_{x} ∑_{i=1}^{n} E[1_{x=x_{u}}(D^{b_{i}}F_{u}(ζ^{b_{1}···b_{i−1}}) − D^{b_{i}}F_{u+v}((ζ^{u,u+v})^{b_{1}···b_{i−1}})

+ (1 − ζ^{b_{1}···b_{n}}(u + v))(F_{u+v}((ζ^{u,u+v})^{b_{1}···b_{n}}) − F_{u}(ζ^{b_{1}···b_{n}})))]

≤ ∑_{x} ∑_{i=1}^{n} \sqrt{P [x = x_{u}]}\{\sqrt{E[(D^{b_{i}}F_{u}(ζ^{b_{1}···b_{i−1}}))^{2}]} + \sqrt{E[(D^{b_{i+1}}F_{u+v}((ζ^{u,u+v})^{b_{1}···b_{i}}))^{2}]}

+ \sqrt{E[(1 − ζ^{b_{1}···b_{n}}(u + v))(F_{u+v}((ζ^{u,u+v})^{b_{1}···b_{n}}) − F_{u}(ζ^{b_{1}···b_{n}})^{2}]}\}

≤ ∑_{|x|=M} \sqrt{P [x = x_{u}]}\{CM^{\frac{1}{2}} \sqrt{M^{d}E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u}(ζ))^{2}]}

+ CM^{\frac{1}{2}} \sqrt{M^{d}E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u+v}(ζ))^{2}]}

+ \sqrt{M^{d}E[(1 − ζ(u + v))(F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ)^{2}]}\}

≤ C ∑_{M} M^{2d− \frac{1}{2}} ρ^{− \frac{(M−1)^{d}}{2}} \{\sqrt{E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u}(ζ))^{2}]} + \sqrt{E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u+v}(ζ))^{2}]}

+ \sqrt{E[(1 − ζ(u + v))(F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ))^{2}]}\}.

Therefore

N^{2} ∑_{u,u+v}(E[(F_{u+v} − F_{u})])^{2}

≤ CN^{2} ∑_{u,u+v}(∑_{M} M^{2d− \frac{1}{2}} ρ^{− \frac{(M−1)^{d}}{2}} \{\sqrt{E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u}(ζ))^{2}]}

+ \sqrt{E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u+v}(ζ))^{2}]} + \sqrt{E[(1 − ζ(u + v))(F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ))^{2}]}\})^{2}

≤ CN^{2} ∑_{u,u+v} 3 ∑_{M} \frac{M^{2d− \frac{1}{2}} ρ^{− \frac{(M−1)^{d}}{2}}}{∑_{m} m^{2d− \frac{1}{2}} ρ^{− \frac{(m−1)^{d}}{2}}} \{E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u}(ζ))^{2}]

+ E[ ∑_{b∈B_{M} (u)}(D^{b}F_{u+v}(ζ))^{2}] + E[(1 − ζ(u + v))(F_{u+v}(ζ^{u,u+v}) − F_{u}(ζ))^{2}]}

≤ CN^{2}E[ ∑_{u,u+v}p(v)( ∑_{y≠u,u+v}(D^{u,u+v}F_{y})^{2}+2(1−ζ(u+v))(F_{u+v}(ζ^{u,u+v})−F_{u}(ζ))^{2})],

as desired. □

9. Exactness of D^{0,v}

In this section we consider the space (X, π_{ρ_{β}, ρ_{ω}}) defined in the preamble to Theorem 0.4. A chain of bonds b_{1}, …, b_{n}; b_{i} = u_{i}, u_{i} + v_{i} , u_{i} ∈ Z^{d}, p(v_{i}) > 0 will be called closed if η^{b_{1}···b_{n}} = η for all η ∈ X. For any function F : X → R and any closed chain of bonds

∑_{i=1}^{n} D^{u_{i},u_{i}+v_{i}}F (η^{b_{1}···b_{i−1}}) = 0.

Let g be a cylinder function. Since D^{u_{i},u_{i}+v_{i}} ∑_{u} τ^{u}g = D^{u_{i},u_{i}+v_{i}} ∑_{u∈B_{l}} τ^{u}g for some large box B_{l} we also have

∑_{i=1}^{n} τ^{u_{i}}f_{g}^{v_{i}}(η^{b_{1}···b_{i−1}}) = ∑_{i=1}^{n} D^{u_{i},u_{i}+v_{i}} ∑_{u} τ^{u}g(η^{b_{1}···b_{i−1}}) = 0.  (9.1)

Definition 2. A function f : X → R^{S} is closed if for every closed chain of bonds

∑_{i=1}^{n} τ^{u_{i}}f^{v_{i}}(η^{b_{1}···b_{i−1}}) = 0.

From (0.16) we see that all exact forms are closed. For c and d in R^{d}, the forms v · c(β(v) − β(0)) and v · d(ω(v) − ω(0)) are closed but not exact. Let 1 < p < 2 be fixed and define L^{p} = {f : ∑_{v∈S} Eπ_{ρ_{β}, ρ_{ω}} [| f^{v} |^{p}] < ∞}. Let

C = {f ∈ H ∩ L^{p} : f closed},

and E be the closure of E_{0} in H ∩ L^{p}. The main result of this section is

Theorem 9.1.

T ∩ C = {v · c(β(v) − β(0))} ⊕ {v · d(ω(v) − ω(0))} ⊕ T ∩ E.

Proof. Let f ∈ T ∩ C. Since the weak closure of a linear subspace is the same as the strong closure we need only find c, d and a sequence g_{n} of cylinder functions such that

v · c(β(v) − β(0)) + v · d(ω(v) − ω(0)) + D^{0,v} ∑_{u} τ^{u}g_{n} → f^{v}

weakly in L^{p} and H. For any set of sites A let F_{A} be the σ–field generated by {η(u), u ∈ A}. Let f_{2N}^{v} = E[f^{v} | F_{B_{2N} (0)}]. Then f_{2N} ∈ T and for any closed chain of bonds in B_{2N}(0),

∑_{i=1}^{n} τ^{u_{i}}f_{2N}^{v_{i}}(η^{b_{1}···b_{i−1}}) = 0.

Therefore there exists g_{2N} of mean 0 on B_{2N}(0) with D^{u,u+v}g_{2N} = τ^{u}f_{2N}^{v} for u, u + v in B_{2N}(0). Since D^{u,u+v} maps T into itself and T^{⊥} into itself, we must have g_{2N} ∈ T . If we let f̃_{2N}^{v} = f_{2N}^{v}1_{X_{2N} \A_{2N}} and g̃_{2N} = g_{2N}1_{X_{2N} \A_{2N}} we still have D^{u,u+v}g̃_{2N} = τ^{u}f̃_{2N}^{v} and g̃_{2N} ∈ T . Then we can apply the spectral gap estimate of Section to get:

E[(g̃_{2N})^{2}] ≤ CN^{2}E[ ∑_{u,u+v} p(v)a(u, u + v)(τ^{u}f̃_{2N}^{v})^{2}]
≤ CN^{d+2}.  (9.2)

For N ≤ m ≤ 2N let g_{m} = E[g̃_{2N} | F_{B_{m}(0)}]. We claim that there exists m_{N}, N ≤ m_{N} ≤ 2N such that the boundary term

\frac{1}{N^{d−1}} ∑_{u+v∉B_{m_{N}} (0)^{u∈Bm_{N} (0)}} E[(D^{u,u+v}g_{m_{N}})^{2}] = O(N^{2}).  (9.3)

To prove this: For any g ∈ L^{2} and set of sites A let g^{A} = E[g | F_{A}]. If u ∈ A and u + v ∉ A then the norms of g^{A} and g^{A\u} are related by a simple formula,

E[(g^{A})^{2}] = E[(g^{A\u})^{2}] + \frac{1}{2}E[(D^{u,u+v}g^{A})^{2}].

Applying this repeatedly we get

B^{d}E[g_{2N}^{2}] = B^{d}E[g_{2N−B}^{2}] + \frac{1}{2} ∑_{u+v∉B_{2N} (0)^{u∈B_{2N} (0)}} E[(D^{u,u+v}g_{2N})^{2}]

where B =| S |. Now applying this repeatedly we get

B^{d}E[g_{2N}^{2}] = B^{d}E[g_{N}^{2}] + ∑_{i=1}^{N/B} \frac{1}{2} ∑_{u+v∉B_{N+Bi}(0)^{u∈B_{N+Bi}(0)}} E[(D^{u,u+v}g_{N+Bi})^{2}].

By (0.17) this gives

∑_{i=1}^{N/B} \frac{1}{2} ∑_{u+v∉B_{N+Bi}(0)^{u∈B_{N+Bi}(0)}} E[(D^{u,u+v}g_{N+Bi})^{2}] ≤ O(N^{d+2}).

But there are N/B terms, so one of them must be O(N^{d+1}) which establishes (0.18). Now consider the formal function

ξ_{N} = \frac{1}{(2m_{N} + 1)^{d}} ∑_{u∈Z^{d}} τ^{−u}g_{m_{N}} .

ξ_{N} does not make sense, but D^{0,v}ξ_{N} does. It is

\frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N}} (0)} τ^{−u}D^{u,u+v}g_{m_{N}} .  (9.4)
+\frac{1}{(2m_{N} + 1)^{d}} ∑_{u+v∉B_{m_{N}} (0)^{u∈Bm_{N} (0)}} τ^{−u}D^{u,u+v}g_{m_{N}} .  (9.5)

First we deal with (0.19). It is equal to

\frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N}} (0)} τ^{−u}E[τ^{u}f̃_{2N}^{v} | F_{B_{m_{N}} (0)}]

= \frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N}} (0)} E[f^{v}1_{X_{2N} \A_{2N}} | F_{B_{m_{N}} (u)}]

= \frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N} −Nϵ}(0)} E[f^{v} | F_{B_{m_{N}} (u)}]  (9.6)
−\frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N} −Nϵ}(0)} E[f^{v}1_{A_{2N}} | F_{B_{m_{N}} (u)}]  (9.7)
+\frac{1}{(2m_{N} + 1)^{d}} ∑_{u+v∈B_{m_{N}} (0)\B_{m_{N} −Nϵ}(0)^{u∈B_{m_{N} −Nϵ}(0)}} E[f^{v}1_{X_{2N} \A_{2N}} | F_{B_{m_{N}} (u)}]  (9.8)

The third term, (0.23), is clearly O(m_{N}^{−1}). We can estimate (0.22) as follows:

E[| \frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N} −Nϵ}(0)} E[f^{v}1_{A_{2N}} | F_{B_{m_{N}} (u)}] |^{p}]

≤ \frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N} −Nϵ}(0)} E[| f^{v}1_{A_{2N}} |^{p}]

= E[| f^{v}1_{A_{2N}} |^{p}] → 0

since P [A_{2N}] → 0. Now we consider (0.21). Since f^{v} ∈ L^{p} for some p > 1 and B_{Nϵ}(0) ⊂ ⋂_{u∈B_{m_{N} −Nϵ}(0)} B_{m_{N}} (u)

\frac{1}{(2m_{N} + 1)^{d}} ∑_{u,u+v∈B_{m_{N} −Nϵ}(0)} E[f^{v} | F_{B_{m_{N}} (u)}] → f^{v}

in L^{p} by the martingale convergence theorem. Thus we have shown that (0.19) converges to f^{v} in L^{p}. Replacing f^{v} by a(0, v)f^{v} and p by 2 in the above proof we see that (0.19) also converges to f^{v} in H.

Finally, we deal with (0.20);

D^{0,v} \frac{1}{(2m_{N} + 1)^{d}} ∑_{u+v∉B_{m_{N}} (0)^{u∈Bm_{N} (0)}} τ^{−u}g_{m_{N}} =^{def} D^{0,v}G_{N}.

By (0.18) it is bounded in L^{2}, so there is a weakly convergent subsequence. We can write

D^{0,v}G_{N} = (β(v) − β(0))G_{N}^{β,v} + (ω(v) − ω(0))G_{N}^{ω,v}

where G_{N}^{β,v} and G_{M}^{ω,v} do not depend on η(0) or η(v). Then

(β(v) − β(0))G_{N}^{β,v} + (ω(v) − ω(0))G_{N}^{ω,v} → (β(v) − β(0))G^{β,v} + (ω(v) − ω(0))G^{ω,v}

weakly in L^{2}. Therefore G_{N}^{β,v} → G^{β,v} and G_{N}^{ω,v} → G^{ω,v} weakly in L^{2}. Suppose that u′, u′ + v′ are not equal to 0 or v.

D^{u′,u′+v′}D^{0,v}G_{N} = D^{0,v}D^{u′,u′+v′}G_{N}
= D^{0,v} \frac{1}{(2m_{N} + 1)^{d}} ∑_{u+v∉B_{m_{N}} (0)^{u∈Bm_{N} (0)}} τ^{−u}D^{u′+u,u′+v′+u}g_{m_{N}}
= D^{0,v} \frac{1}{(2m_{N} + 1)^{d}} ∑_{u+v∉B_{m_{N}} (0)^{u∈Bm_{N} (0)}} τ^{−u}E[τ^{u′+u}f̃_{2N}^{v′} | F_{B_{m_{N}} (0)}]

→ 0 in L^{p} as N → ∞. Thus G^{β,v} and G^{ω,v} are constants. Since L^{p} ∩ H is weaker than L^{2} we have shown that (0.20) converges weakly in L^{p} ∩ H to G^{β,v}(β(v) − β(0)) + G^{ω,v}(ω(v) − ω(0)) for some constants G^{β,v} and G^{ω,v}. This means that we have found cylinder functions g_{N} such that

D^{0,v} ∑_{u} τ^{u}g_{N} →^{w−L^{p}∩H} f^{v} + G^{β,v}(β(v) − β(0)) + G^{ω,v}(ω(v) − ω(0)).

But D^{0,v} ∑_{u} τ^{u}g_{N} ∈ C and G^{β,v}(β(v) − β(0)) + G^{ω,v}(ω(v) − ω(0)) is only in C if G^{β,v} = v · c and G^{ω,v} = v · d for some c and d in R^{d}. This proves the theorem. □

10. Central limit theorem variances I

For ϕ and ψ in R^{d} let

ν_{i}(u, ϕ, ψ) = ϕ_{i}(β(u + e_{i}) − β(u)) + ψ_{i}(ω(u + e_{i}) − ω(u)),  (10.1)

and ν(u, ϕ, ψ) = ∑_{i=1}^{d} ν_{i}(u, ϕ, ψ). Write Φ = (ϕ, ψ). For the discrete points ρ_{β}, ρ_{ω} = \frac{n}{(2k+1)^{d}} , \frac{m}{(2k+1)^{d}}; n and m integers, and ρ_{β} + ρ_{ω} < 1 define (Φ, B_{k}(ρ_{β}, ρ_{ω})Φ) to be the central limit theorem variance of V in the stationary process on the hyperplane with initial measure u_{ρ_{β}, ρ_{ω}}^{k}.

(Φ, B_{k}(ρ_{β}, ρ_{ω})Φ) = \frac{1}{(2(k−1)+1)^{d}} lim_{t→∞} E[(\frac{1}{\sqrt{t}} ∫_{0}^{t} ∑_{u∈B_{k−1}(0)} ν(u, Φ)ds)^{2}]

Define, for sequences ρ_{β}^{k}, ρ_{ω}^{k} = \frac{n^{k}}{(2k+1)^{d}} , \frac{m^{k}}{(2k+1)^{d}},

(Φ, B(ρ_{β}, ρ_{ω})Φ) = \overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}}(Φ, B_{k}(ρ_{β}^{k}, ρ_{ω}^{k})Φ).

Theorem 10.1.

(Φ, B(ρ_{β}, ρ_{ω})Φ) ≤ 4(Φ, (h′′Ah′′(ρ_{β}, ρ_{ω}))^{−1}Φ).

Proof. The result can be rephrased as follows: For any M with

\overline{lim}_{k→∞}(Φ, B_{k}(ρ_{β}^{k}, ρ_{ω}^{k})Φ) > M  (10.2)

we want (Φ, (h′′Ah′′)^{−1}Φ) ≥ \frac{M}{4} . We will prove this in the following manner: We will produce an f ∈ T ∩ C satisfying

Eπ_{ρ_{β}, ρ_{ω}} [∑_{i=1}^{d} f^{e_{i}}(ϕ_{i}(β(e_{i}) − β(0)) + ψ_{i}(ω(e_{i}) − ω(0)))] = 2,  (10.3)
Eπ_{ρ_{β}, ρ_{ω}} [∑_{v} p(v)a(0, v)(f^{v})^{2}] ≤ 4M^{−1}.  (10.4)

By Theorem 0.7, f^{v} is the limit in L^{p} ∩ H of

v · c(β(v) − β(0)) + v · d(ω(v) − ω(0)) + D^{0,v} ∑_{u} τ^{u}g_{n}

for some c and d in R^{d} and sequence g_{n} of local functions. Since

Eπ_{ρ_{β}, ρ_{ω}} [(ϕ_{i}(β(e_{i}) − β(0)) − ψ_{i}(ω(e_{i}) − ω(0)))D^{0,v} ∑_{u} τ^{u}g_{n}] = 0

we get

lim_{n→∞} Eπ_{ρ_{β}, ρ_{ω}} [∑_{i=1}^{d}(c_{i}(β(e_{i}) − β(0)) + d_{i}(ω(e_{i}) − ω(0)))

× (ϕ_{i}(β(e_{i}) − β(0)) + ψ_{i}(ω(e_{i}) − ω(0)))]

= (2(h′′)^{−1}(c, d), (ϕ, ψ)) = 2.

Then (0.27) gives

Eπ_{ρ_{β}, ρ_{ω}} [∑_{v} p(v)a(0, v)(v·c(β(v)−β(0))+v·d(ω(v)−ω(0))−D^{0,v} ∑_{u}τ^{u}g)^{2}]≤4M^{−1}.

Therefore

((ϕ, ψ)(h′′Ah′′)^{−1}(ϕ, ψ)) = sup_{x,y∈R^{d}} \frac{((h′′)^{−1}(x, y), (ϕ, ψ))}{((x, y), A(x, y))}
≥ \frac{((h′′)^{−1}(c, d), (ϕ, ψ))}{((c, d), A(c, d))}
≥ \frac{M}{4} .

So the problem is reduced to finding f ∈ T ∩ H satisfying (0.26) and (0.27). By (0.25) there exists a sequence g^{k} such that

(Eu_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k} [g^{k} ∑_{u∈B_{k−1}} ν(u)])^{2} > \frac{M(2k + 1)^{d}}{2} D_{k}(g^{k})

By the remarks preceding Theorem 0.4, g^{k} can be taken to be of the form g^{k} = ∑_{u}(aβ(u) + bω(u))F_{u}(ζ). Summing by parts and using the definition of the Dirichlet form, we have, for the expectation with respect to u_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k},

(E[∑_{i} ∑_{u,u+e_{i}∈B_{k−1}(0)} D^{u,u+e_{i}}g^{k}ν_{i}(u)])^{2}

> M(2k + 1)^{d}E[ ∑_{u,u+v∈B_{k}(0)} p(v)a(u, u + v)(D^{u,u+v}g^{k})^{2}].  (10.5)

Let m < k − 1, 1 < p < 2, and q be the conjugate exponent of p. We claim that there exists a sequence y^{k} ∈ B_{k−1−m}(0) and a constant C_{0} such that

(E[∑_{i} ∑_{u,u+e_{i}∈B_{m}(y^{k})} D^{u,u+e_{i}}g^{k}ν_{i}(u)])^{2}

≥ (1 − C_{0}k^{− \frac{1}{2q}})M(2m + 1)^{d}E[  ∑_{u,u+v∈B_{m}(y^{k})} p(v)a(u, u + v)(D^{u,u+v}g^{k})^{2}]. (10.6)

C_{0} is a fixed constant to be described. Suppose there is no such sequence y^{k}. We can average over y ∈ B_{k−1−m}(0) and use Jensen’s inequality to obtain

(E[\frac{1}{(2(k−1−m)+1)^{d}} ∑_{y∈B_{k−1−m}} \frac{1}{(2m+1)^{d}} ∑_{i} ∑_{u,u+e_{i}∈B_{m}(y^{k})} D^{u,u+e_{i}}g^{k}ν_{i}(u)])^{2}

< (1 − C_{0}k^{− \frac{1}{2q}})\frac{M}{(2k + 1)^{d}} D_{k}(g^{k}).  (10.7)

The missing boundary terms on the left hand side can be estimated by using Lemma 4 and Hölder’s inequality,

E[∑_{i} ∑_{u or u+e_{i}∉B_{k−1−m}^{u,u+e_{i}∈B_{k−1}}} D^{u,u+e_{i}}g^{k}ν_{i}(u)]

≤ C_{1}(E[ ∑_{u,u+v∈B_{k}} p(v) | D^{u,u+v}g^{k} |^{p}])^{\frac{1}{p}} (E[∑_{i} ∑_{u or u+e_{i}∉B_{k−1−m}^{u,u+e_{i}∈Bk−1}} | ν_{i}(u) |^{q}])^{\frac{1}{q}}

≤ C_{2}(k^{d( \frac{2−p}{2} )}(D_{k}(g^{k}))^{\frac{p}{2}} )^{\frac{1}{p}} (mk^{d−1})^{\frac{1}{q}} = C_{2}(D_{k}(g^{k}))^{\frac{1}{2}} k^{\frac{d}{2} − \frac{1}{q}} .  (10.8)

Taking C_{0} > 2C_{2}, (0.30), together with (0.31), contradict (0.28), establishing (0.29). By rescaling the g^{k} we can restate this as

\frac{1}{(2m+1)^{d}} Eu_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k} [∑_{i} ∑_{u,u+e_{i}∈B_{m}(y^{k})} D^{u,u+e_{i}}g^{k}ν_{i}(u)] ≥ 2 + O(k^{− \frac{1}{q}})  (10.9)
\frac{1}{(2m+1)^{d}} Eu_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k} [  ∑_{u,u+v∈B_{m}(y^{k})} p(v)a(u, u + v)(D^{u,u+v}g^{k})^{2}] ≤ 4M^{−1}.  (10.10)

Let

g^{k,m} = τ^{−y}Eu_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k} [g^{k} | F_{B_{m}(y^{k})}]

and let μ^{k} be the projection of u_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k} onto B_{m}(0). We have

\frac{1}{(2m+1)^{d}} E^{μ^{k}}[∑_{i} ∑_{u,u+e_{i}∈B_{m}(0)} D^{u,u+e_{i}}g^{k,m}ν_{i}(u)] ≥ 2 + O(k^{− \frac{1}{q}})  (10.11)
\frac{1}{(2m+1)^{d}} E^{μ^{k}}[ ∑_{u,u+v∈B_{m}(0)} p(v)a(u, u + v)(D^{u,u+v}g^{k,m})^{2}] ≤ 4M^{−1}.  (10.12)

If we replace g^{k,m} by g̃^{k,m} = 1_{X_{k}\A_{m}}g^{k,m} then the second inequality is still true and, using Lemma 4 and Hölder’s inequality, we can show that the error in (0.35) goes to zero as k, then m → ∞. The g̃^{k,m} are still of the form g^{k} = ∑_{u}(aβ(u) + bω(u))F_{u}(ζ). Since we have neutralised overfull configurations by multiplying by 1_{X_{k}\A_{m}} we can apply Lemma 11 to μ^{k}. This gives

E^{μ^{k}}[(g̃^{k,m})^{2}] ≤ Cm^{2}E^{μ^{k}}[ ∑_{u,u+v∈B_{m}(0)} p(v)(D^{u,u+v}g̃^{k,m})^{2}] ≤ 4Cm^{2}M^{−1},

i.e. g̃^{k,m} is pointwise bounded independent of k. So a subsequence g̃^{k,m} → g^{m} with

\frac{1}{(2m+1)^{d}} Eπ_{ρ_{β}, ρ_{ω}} [∑_{i} ∑_{u,u+e_{i}∈B_{m}(0)} D^{u,u+e_{i}}g̃^{m}ν_{i}(u)] ≥ 2  (10.13)
\frac{1}{(2m+1)^{d}} Eπ_{ρ_{β}, ρ_{ω}} [ ∑_{u,u+v∈B_{m}(0)} p(v)a(u, u + v)(D^{u,u+v}g̃^{m})^{2}] ≤ 4M^{−1}.  (10.14)

For c < m let

ξ^{m,c} = \frac{1}{(2m+1)^{d}} ∑_{x∈B_{m−c}(0)} τ^{x}g^{m}.

This has the property that if u ∈ B_{c}(0),

Eπ_{ρ_{β}, ρ_{ω}} [| D^{0,v}ξ^{m,c} − τ^{u}D^{u,u+v}ξ^{m,c} |]

= \frac{1}{(2m+1)^{d}} Eπ_{ρ_{β}, ρ_{ω}} [| D^{0,v}( ∑_{x∈B_{m−c}(0)} τ^{x}g^{m} − ∑_{x∈B_{m−c}(u)} τ^{x}g^{m}) |]

≤ \frac{1}{(2m+1)^{d}} ∑_{u∈B_{m}(0)} Eπ_{ρ_{β}, ρ_{ω}} [| D^{u,u+v}g^{m} | 1_{u∈B_{m−c}(0)△B_{m−c}(u)}]

≤ (\frac{1}{(2m+1)^{d}} ∑_{u∈B_{m}(0)}Eπ_{ρ_{β}, ρ_{ω}} [|D^{u,u+v}g^{m} |^{p}])^{\frac{1}{p}} (\frac{1}{(2m+1)^{d}} ∑_{u∈B_{m}(0)}Eπ_{ρ_{β}, ρ_{ω}} [1_{u∈B_{m−c}(0)△B_{m−c}(u)}])^{\frac{1}{q}}

≤ (C(2m + 1)^{\frac{dp}{2}} (Eπ_{ρ_{β}, ρ_{ω}} [ ∑_{u,u+v∈B_{m}(0)}p(v)(D^{u,u+v}g^{k,m})^{2}])^{\frac{p}{2}} )^{\frac{1}{p}} (Cm^{−1})^{\frac{1}{q}} .  (10.15)

What (0.38) shows is that for any closed chain of bonds inside B_{c}(0)

∑_{i=1}^{n} τ^{u_{i}}D^{0,v_{i}}ξ^{m,c}(η^{b_{1}···b_{i−1}}) = O(m^{− \frac{1}{q}}).

The ξ^{m,c} also satisfy:

Eπ_{ρ_{β}, ρ_{ω}} [∑_{i=1}^{d} D^{0,e_{i}}ξ^{m,c}ν_{i}(0)] ≥ 2 + O(m^{− \frac{1}{q}})  (10.16)

Eπ_{ρ_{β}, ρ_{ω}} [∑_{v} p(v)a(0, v)(D^{0,v}ξ^{m,c})^{2}] ≤ 4M^{−1}  (10.17)

The second of these follows from Jensen’s inequality. The first differs from (0.36) by

\frac{1}{(2m+1)^{d}} Eπ_{ρ_{β}, ρ_{ω}} [ ∑_{u+e_{i}∈B_{m}(0)^{u∈Bm(0)\B_{m−c}(0)}} D^{u,u+e_{i}}g̃^{m}ν_{i}(u)] = O(m^{− \frac{1}{q}})

Write f^{m,c,v} = D^{0,v}ξ^{m,c}.

Eπ_{ρ_{β}, ρ_{ω}} [| f^{m,c,v} |^{p}] = Eπ_{ρ_{β}, ρ_{ω}} [| \frac{1}{(2m+1)^{d}} ∑_{x∈B_{m−c}(0)} τ^{x}D^{−x,−x+v}g^{m} |^{p}]

\frac{1}{(2m+1)^{d}} ∑_{x∈B_{m−c}(0)} Eπ_{ρ_{β}, ρ_{ω}} [| D^{−x,−x+v}g^{m} |^{p}] < C

So we can take a subsequence which converges to f^{v} weakly in L^{p} as m, then c → ∞. From (0.38), (0.39), (0.40) we get that f ∈ T ∩C and satisfies (0.26) and (0.27), which completes the proof. □

11. Central limit theorem variances II

For notational purposes let F (ρ_{β}, ρ_{ω}) ∈ R^{d} × R^{d} be given by

F = (F β, F ω) = \frac{1}{2}(h′′A) ( \begin{matrix}ϕ \\ ψ \end{matrix} ) .  (11.1)

Let V^{k} = V_{I}^{k} − V_{II}^{k} where, for a cylinder function g,

V_{I}^{k}(u) = ∑_{v} p(v)(v · ϕ\frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}(u)} a(y, y + v)(β(y + v) − β(y))

+ v · ψ \frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}(u)} a(y, y + v)(ω(y + v) − ω(y)) − L̂_{2k+1}g, (11.2)

and, for ν defined in (0.24),

V_{II}^{k}(u) = \frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}(u)} ν(y, F (ρ_{β}^{k}(u), ρ_{ω}^{k}(u)).  (11.3)

Lemma 12. For the expectation with respect to the non–speeded up process on B_{k} with initial distribution u_{ρ_{β}^{k}, ρ_{ω}^{k}}^{k},

\overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}} \frac{1}{(2k+1)^{d}} lim_{t→∞} E[(\frac{1}{\sqrt{t}} ∫_{0}^{t} V^{k}(η_{s})ds)^{2}] ≤ Q_{g}((ϕ, ψ)) − ((ϕ, ψ), A(ϕ, ψ)).

Proof.

lim_{t→∞} E[(\frac{1}{\sqrt{t}} ∫_{0}^{t} V^{k}(η_{s})ds)^{2}] = 2 < V^{k}, (L̂_{2k+1})^{−1}V^{k} >

= 2 <V_{I}^{k}, (L̂_{2k+1})^{−1}V_{I}^{k} >−4 <V_{II}^{k}, (L̂_{2k+1})^{−1}V_{I}^{k} >+2 <V_{II}^{k}, (L̂_{2k+1})^{−1}V_{II}^{k} > . (11.4)

We have already calculated in Section that

\overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}} \frac{1}{(2k + 1)^{d}} 2 < V_{II}^{k}, (L̂_{2k+1})^{−1}V_{II}^{k} >≤ ((ϕ, ψ), A(ϕ, ψ)).  (11.5)

We now calculate the two other terms. Notice that

V_{I} = L̂_{2k+1}[ ∑_{x∈B_{k}(0)} x · ϕβ(x) + x · ψω(x) + ∑_{x∈B_{k−l}(0)} τ^{x}g]

so that

2 < V_{I}^{k}, (L̂_{2k+1})^{−1}V_{I}^{k} >

= E[ ∑_{u,u+v∈B_{k}(0)} p(v)a(u, u + v)(D^{u,u+v}[ ∑_{x∈B_{k}(0)} x · ϕβ(x) + x · ψω(x) + ∑_{x∈B_{k−l}(0)}τ^{x}g])^{2}]

= E[ ∑_{u,u+v∈B_{k}(0)} p(v)a(u, u+v)(v·ϕ(β(u+v)−β(u))+v·ψ(ω(u+v)−ω(u))+D^{u,u+v} ∑_{x∈B_{k−l}(0)}τ^{x}g)^{2}]

= E[ ∑_{u,u+v∈B_{k}(0)} p(v)a(0, v)(v·ϕ(β(v)−β(0))+v·ψ(ω(v)−ω(0))+D^{0,v} ∑_{x∈B_{k−l}(u)}τ^{x}g)^{2}]

= E[ ∑_{u,u+v∈B_{k−2l}(0)} p(v)a(0, v)(v·ϕ(β(v)−β(0))+v·ψ(ω(v)−ω(0))+D^{0,v} ∑_{x∈B_{k−l}(u)}τ^{x}g)^{2}]

+E[ ∑_{u or u+v∉B_{k−2l}(0)^{u,u+v∈B_{k}(0)}} p(v)a(0, v)(v·ϕ(β(v)−β(0))+v·ψ(ω(v)−ω(0))+D^{0,v} ∑_{x∈B_{k−l}(u)}τ^{x}g)^{2}].

The second term is a sum of k^{d−1}2l terms each of order l^{d}, so in the limit, when divided by k^{d}, it is negligible. If u ∈ B_{k−2l}(0) then

D^{0,v} ∑_{x∈B_{k−l}(u)} τ^{x}g = D^{0,v} ∑_{x∈B_{k−l}(0)} τ^{x}g

so in the first term all the summands are the same. We get

\overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}} \frac{1}{(2k + 1)^{d}} 2 < V_{I}^{k}, (L̂_{2k+1})^{−1}V_{I}^{k} >= Q_{g}((ϕ, ψ)).  (11.6)

It remains to calculate the cross term

\frac{1}{(2k + 1)^{d}} 4 < V_{II}^{k}, (L̂_{2k+1})^{−1}V_{I}^{k} > .

= \frac{4}{(2k + 1)^{d}} E[( ∑_{y∈B_{k}} ∑_{i=1}^{d} F β_{i}(β(u + e_{i}) − β(u)) + F ω_{i}(ω(u + e_{i}) − ω(u)))

× ( ∑_{x∈B_{k}} x · ϕβ(x) + x · ϕω(x) + τ^{x}g)]

= 2 ∑_{i=1}^{d} \frac{1}{(2k + 1)^{d}} ∑_{y∈B_{k}} E[(F β_{i}(β(u + e_{i}) − β(u)) + F ω_{i}(ω(u + e_{i}) − ω(u)))

× (ϕ_{i}(β(u + e_{i}) − β(u)) + ψ_{i}(ω(u + e_{i}) − ω(u)) + D^{u,u+e_{i}} ∑_{x∈B_{k}} τ^{x}g)].

Since E[(β(u + e_{i}) − β(u)) ∑_{x∈B_{k−l_{0}}} τ^{x}g] = 0 we get

\overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}} \frac{1}{(2k + 1)^{d}} 2 < V_{II}^{k}, (L̂_{2k+1})^{−1}V_{I}^{k} >= 2((ϕ, ψ), A(ϕ, ψ)).  (11.7)

The proof is completed by plugging (0.45), (0.46), and (0.47) into (0.44). □

We would like to apply Theorem 0.6 using the results of Lemma 12. Unfortunately, V_{II}^{k} does not satisfy the condition that L_{N} + ∑_{u} V_{II}^{k}(u) have principal eigenvalue bounded by BN^{d}. Therefore we introduce cutoff functions. Recall that A_{l} is the set of configurations having one or fewer occupied sites in the box B_{l}. Define

ν^{l}(u, ϕ, ψ) = ∑_{i=1}^{d} ϕ_{i}(β(u + e_{i})τ^{u+e_{i}}1_{X_{N} \A_{l}} − β(u)τ^{u}1_{X_{N} \A_{l}})

+ ψ_{i}(ω(u + e_{i})τ^{u+e_{i}}1_{X_{N} \A_{l}} − ω(u)τ^{u}1_{X_{N} \A_{l}})  (11.8)

and

V_{II}^{k,l}(u). = \frac{1}{(2k−l)^{d}} ∑_{y∈B_{k−l}(u)} ν^{l}(y, F (ρ_{β}^{k}(u), ρ_{ω}^{k}(u))  (11.9)

Using Lemma 6 we can show that for ρ < 1 there exists a constant C depending only on ρ such that

\overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}} \frac{1}{(2k+1)^{d}} < (V_{II}^{k,l}−V_{II}^{k}), (L̂_{2k+1})^{−1}(V_{II}^{k,l}−V_{II}^{k}) >≤ Cl^{2d( \frac{p−1}{p})}ρ^{\frac{l^{d}}{2}} , (11.10)

so that for large l, Lemma 12 is essentially unchanged by substituting V_{II}^{k,l} for V_{II}^{k}. Now let V_{l}(u) = \frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−l−1}} ν^{l}(y, ϕ(u/N), ψ(u/N)), and let λ_{N}^{l} be the principal eigenvalue of L_{N} + N ∑_{u} V_{l}(u). Using Lemma 5, we can show that there exists a constant B such that

λ_{N}^{l} ≤ Bl^{4d}N^{d}.  (11.11)

Therefore, for each l our cutoff functions satisfy the conditions of Theorem 0.6. The next result is needed in the proof of Lemma 13, but has independent interest.

Theorem 11.1. The self-diffusion covariance D_{s}(ρ) is continuous for 0 < ρ < 1.

Proof. We actually prove that (Φ, AΦ) is continuous in ρ_{β} and ρ_{ω}. Since (Φ, AΦ) is the infimum over cylinder functions g of Q_{g}(Φ), it is upper semicontinuous. In Section we showed that

4(Φ, (h′′Ah′′)^{−1}(ρ_{β}, ρ_{ω})Φ) ≥ \overline{lim}_{ρ_{β}^{k},ρ_{ω}^{k}→ρ_{β},ρ_{ω}^{k→∞}}(Φ, B_{k}(ρ_{β}^{k}, ρ_{ω}^{k})Φ).

where B_{k} are continuous. By Lemma 12 we have equality here. Therefore (Φ, (h′′Ah′′)^{−1}Φ) is upper semicontinuous. But (h′′)^{−1} is continuous so (Φ, A^{−1}Φ) is upper semicontinuous. Then (Φ, AΦ) = sup_{x}(Φ, x) − (x, A^{−1}x) is lower semicontinuous. □

Define

Ṽ_{II}^{k,l}(u) = \frac{1}{2Nϵ+1} ∑_{|h|≤Nϵ^{x=u+he_{i}}} \frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}(x)} ν^{l}(y, F (ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u)).  (11.12)

Lemma 13.

\overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{1}{N^{d}} log E^{P_{N}} [exp N^{d}G_{N,ϵ,k,m,l}] ≤ 0

where

G_{n,ϵ,k,m,l} = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} N(V_{II}^{k,l} − Ṽ_{II}^{k,l})(\frac{u}{N} , s, \{η(x)\}_{x∈B_{Nϵ}(u)}) − mτ^{u}1_{A_{k}}ds.

Proof. We can rearrange ∑_{u} N(V_{II}^{k,l} − Ṽ_{II}^{k,l}) to get ∑_{u} ν^{l}(u, J(u)) where J ∈ R^{2d} is given by

J(u) = \frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}(u)}\frac{1}{2Nϵ+1} ∑_{|h|≤Nϵ^{x=y+he_{i}}}(F (\frac{x}{N} , s, ρ_{β}^{Nϵ}(x), ρ_{ω}^{Nϵ}(x)) − F (\frac{u}{N} , s, ρ_{β}^{k}(u), ρ_{ω}^{k}(u))).

By Feynman-Kac formula and spectral theorem

log E^{P_{N}} [exp(∫_{0}^{T} N ∑_{u} ν^{l}(u, J(u)) − mτ^{u}1_{A_{k}}ds)]

is majorized by

sup_{f_{N}} Eπ^{N} [f_{N}N ∑_{u} aν^{l}(u, J(u)) − mτ^{u}1_{A_{k}}] − N^{2}D_{N}(\sqrt{f_{n}}).

The functional J(u) only involves empirical densities ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u), ρ_{β}^{k}(u) and ρ_{ω}^{k}(u). Since N ≫ k ≫ l, J(u) is unaffected by any exchange between sites in B_{2l}(u). Therefore we can apply Lemma 5 to obtain

E[f_{N}ν^{l}(u, J(u)) ≤ Cl^{2d} ∑_{y,y+v∈B_{2l}(u)}\sqrt{E[a(y, y + v)(D^{y,y+v}(\sqrt{f_{n}}))^{2}]}\sqrt{E[| J(u) |^{2} f_{N}]}.

Then we can estimate the eigenvalue

sup_{f_{N}} Eπ^{N} [f_{N}N ∑_{u} aν^{l}(u, J(u)) − mτ^{u}1_{A_{k}}] − N^{2}D_{N}(\sqrt{f_{n}}).

≤ sup_{f_{N}} ∑_{u} CNl^{2d} ∑_{y,y+v∈B_{2l}(u)}\sqrt{Eπ^{N}[a(y, y+v)(D^{y,y+v}(\sqrt{f_{n}}))^{2}]}\sqrt{Eπ^{N}[| J(u) |^{2} f_{N}]}

− \frac{N^{2}}{2(2l + 1)^{d}} Eπ^{N} [a(y, y + v)(D^{y,y+v}(\sqrt{f_{n}}))^{2}] − \frac{N^{2}}{2} D_{N}(\sqrt{f_{n}}) − mEπ^{N} [τ^{u}1_{A_{k}}]

≤ sup_{f_{N}} ∑_{u} Cl^{6d}E^{f_{N}π^{N}} [| J(u) |^{2} −mτ^{u}1_{A_{k}}] − \frac{N^{2}}{2} D_{N}(\sqrt{f_{n}}).

Now by Theorem 0.9, F is uniformly continuous so we can apply Lemma 7 to get

\overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{1}{N^{d}} sup_{f_{N}} ∑_{u}Cl^{6d}E^{f_{N}π^{N}}[|J(u)|^{2}−mτ^{u}1_{A_{k}}]− \frac{N^{2}}{2} D_{N}(\sqrt{f_{n}})≤0

as desired. □

12. Hydrodynamic limit

In this section we apply the variance calculations of Sections and and Theorem 0.6 to prove the hydrodynamic limit.

Let Q̃ be any limit point of Q̃_{N} (cf.p.9). We first prove a preliminary estimate. Lemma 14.

EQ̃[∫_{T^{d}} ∫_{0}^{∞}(∇ ( \begin{matrix}λβ \\ λω \end{matrix} ) , A∇ ( \begin{matrix}λβ \\ λω \end{matrix} ))dtdθ] ≤ 8 log 3.

Proof. We first remark that μβ, μω have densities ρ_{β}, ρ_{ω} almost surely with respect to Q̃, so the integrand above is well defined. For test functions ϕ, ψ : T^{d} → R^{d} and ν^{l} as in (0.48) define the functional

F_{N,ϵ,k,m,l}(ϕ, ψ) = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} Nν^{l}(u, ϕ(\frac{u}{N}), ψ(\frac{u}{N}))

− 2((ϕ(\frac{u}{N}), ψ(\frac{u}{N})), (h′′Ah′′)^{−1}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(ϕ(\frac{u}{N}), ψ(\frac{u}{N})))

− Cl^{2d( \frac{p−1}{p})}(ρ^{Nϵ}(u))^{\frac{l^{d}}{2}} − mτ^{u}1_{A_{k}}ds. Let {ϕ^{i}}_{i=1,2,...}, {ψ^{j}}_{j=1,2...} be countable dense subsets of C^{∞}(T^{d} → R^{d}). By The-

orem 0.8, (0.50), (0.51), Theorem 0.6, and the entropy inequality,

\overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} EQ̃_{N} [max_{i,j≤M} F_{N,ϵ,k,m,l}(ϕ^{i}, ψ^{j})]

≤ \overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{1}{N^{d}} log ∑_{i,j≤M}EP̃_{N} [exp N^{d}F_{N,ϵ,k,m,l}(ϕ^{i}, ψ^{j})] + \frac{H(Q_{N}; P_{N})}{N^{d}}

≤ log 3.

We write F_{N,ϵ,k,m,l} = F_{N,ϵ}^{1} + F_{N,ϵ,k,m,l}^{2} where

F_{N,ϵ}^{1} =∫_{0}^{T}\frac{1}{N^{d}} ∑_{u} N(∑_{i=1}^{d}(ϕ_{i}(\frac{u−e_{i}}{N})−ϕ_{i}(\frac{u}{N}))β(u) + (ψ_{i}(\frac{u−e_{i}}{N})−ψ_{i}(\frac{u}{N}))ω(u))

− 2((ϕ(\frac{u}{N}), ψ(\frac{u}{N})), (h′′Ah′′)^{−1}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(ϕ(\frac{u}{N}), ψ(\frac{u}{N})))ds,

and

F_{N,ϵ,k,m,l}^{2} =∫_{0}^{T}\frac{1}{N^{d}} ∑_{u}N(∑_{i=1}^{d}(ϕ_{i}(\frac{u−e_{i}}{N})−ϕ_{i}(\frac{u}{N}))β(u)+(ψ_{i}(\frac{u−e_{i}}{N})−ψ_{i}(\frac{u}{N}))ω(u))τ^{u}1_{A_{l}}

− Cl^{2d( \frac{p−1}{p})}(ρ^{Nϵ}(u))^{\frac{l^{d}}{2}} − mτ^{u}1_{A_{k}}ds.

By (0.14),

\overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} EQ̃_{N} [max_{i,j≤M} F_{N,ϵ,k,m,l}^{2}] = 0.

Therefore

\overline{lim}_{ϵ→0} \overline{lim}_{N→∞} EQ̃_{N} [max_{i,j≤M} F_{N,ϵ}^{1}] ≤ log 3.

As functionals F_{N,ϵ}^{1} converge, as N → ∞, to

F_{ϵ}^{1} = ∫_{T^{d}} ∫_{0}^{T} −∇ · ϕ(θ)ρ_{β}(θ) − ∇ · ψ(θ)ρ_{ω}(θ)

− 2((ϕ(θ), ψ(θ))(h′′Ah′′)^{−1}(\frac{μβ(θ − ϵ, θ + ϵ)}{2ϵ}, \frac{μω(θ − ϵ, θ + ϵ)}{2ϵ})(ϕ(θ), ψ(θ)))dsdθ,

and the convergence is uniform, so

\overline{lim}_{ϵ→0} EQ̃[max_{i,j≤M} F_{ϵ}^{1}] ≤ log 3.

Now there is a fixed δ > 0 such that ρ(θ) < 1 − δ almost everywhere with respect to Q̃ and the function ρ_{β}, ρ_{ω} → ((ϕ, ψ), (h′′Ah′′)^{−1}(ρ_{β}, ρ_{ω})(ϕ, ψ)) is continuous away from ρ = 1. Therefore F_{ϵ}^{1} is uniformly bounded Q̃–a.e., and this allows us to take the limit inside the expectation. Then we can let M → ∞ to get

EQ̃[sup_{ϕ,ψ} ∫_{T^{d}} ∫_{0}^{T} −∇·ϕρ_{β} −∇·ψρ_{ω}−2((ϕ, ψ), (h′′Ah′′)^{−1}(ρ_{β}, ρ_{ω})(ϕ, ψ))dsdθ] ≤ log 3.

Since h′′∇ ( \begin{matrix}ρ_{β} \\ ρ_{ω} \end{matrix} ) = ∇ ( \begin{matrix}λβ \\ λω \end{matrix} ) this is the variational form of the lemma. □ The main result is

Theorem 12.1. For test functions ϕ, ψ : T^{d} → R the following holds Q̃–a.e.

∫_{T^{d}} ϕ(θ)ρ_{β}(θ, T) + ψ(θ)ρ_{ω}(θ, t)dθ − ∫_{T^{d}} ϕ(θ)ρ_{β}(θ, 0) + ψ(θ)ρ_{ω}(θ, 0)dθ

= \frac{1}{2} ∫_{T^{d}} ∫_{0}^{T} ((∇ϕ(θ), ∇ψ(θ)), (Ah′′)(ρ_{β}(θ, t), ρ_{ω}(θ, t))(∇ρ_{β}(θ, t), ∇ρ_{ω}(θ, t))dtdθ.

Proof: By (0.13) we have that Q̃–a.e.

∫_{T^{d}} ϕ(θ)ρ_{β}(θ, T) + ψ(θ)ρ_{ω}(θ, T)dθ − ∫_{T^{d}} ϕ(θ)ρ_{β}(θ, 0) + ψ(θ)ρ_{ω}(θ, 0)dθ
=< (∇ϕ, γβ(T)) > + < (∇ψ, γω(T)) > .  (12.1)

For test functions ϕ, ψ : T^{d} → R^{d} and V_{I}^{k}, Ṽ_{II}^{k,l} as in (0.42), (0.52) , and Q_{g} as in (0.9) consider the functional

G_{N,ϵ,k,m,l}(ϕ, ψ, g) = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} N(V_{I}^{k}(u, g, ϕ(\frac{u}{N}), ψ(\frac{u}{N})) − Ṽ_{II}^{k,l}(u, ϕ(\frac{u}{N}), ψ(\frac{u}{N})))

− (Q_{g}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u), ϕ(\frac{u}{N}), ψ(\frac{u}{N}))

− ((ϕ(\frac{u}{N}), ψ(\frac{u}{N})), A(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(ϕ(\frac{u}{N}), ψ(\frac{u}{N})))

− Cl^{2d( \frac{p−1}{p})}(ρ^{Nϵ}(u))^{\frac{l^{d}}{2}} − mτ^{u}1_{A_{k}}dt.

Let {ϕ^{i}}_{i=1,2,...}, {ψ^{j}}_{j=1,2...} be countable dense subsets of C^{∞}(T^{d} → R^{d}) and {g^{h}}_{h=1,2,...} an enumeration of the cylinder functions on X. By Lemmas 12, 13, (0.50), (0.51), and a slight modification of Theorem 0.6,

\overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} EQ̃_{N} [ max_{i,j,h≤M} G_{N,ϵ,k,m,l}(ϕ^{i}, ψ^{j}, g^{h})]

≤ \overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} \frac{1}{N^{d}} log ∑_{i,j,h≤M}EP̃_{N} [exp N^{d}G_{N,ϵ,k,m,l}(ϕ^{i}, ψ^{j}, g^{h})]

+ \frac{H(Q_{N}; P_{N})}{N^{d}}

≤ log 3.

We can write G_{N,ϵ,k,m,l} = G_{N,ϵ,k}^{1} + G_{N}^{2} + G_{N,ϵ,k,m,l}^{3} + G_{N,ϵ}^{4} where

G_{N,ϵ,k}^{1} = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u,v} p(v)(\frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}(u)}a(y, y+v)(v·ϕ(\frac{u}{N})(β(y+v) − β(y))

+v·ψ(\frac{u}{N})(ω(y+v) − ω(y)))

− ∑_{i=1}^{d} F β_{i}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(\frac{ρ_{β}^{k}(u + Nϵe_{i}) − ρ_{β}^{k}(u − Nϵe_{i})}{2Nϵ + 1})

+ F ω_{i}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(\frac{ρ_{ω}^{k}(u + Nϵe_{i}) − ρ_{ω}^{k}(u − Nϵe_{i})}{2Nϵ + 1}),

G_{N}^{2} = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} NL̂_{N}g(\frac{u}{N} , t, \{η(x)\}_{x∈B_{l_{0}}(u)})dt,

G_{N,ϵ,k,m,l}^{3} = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} −Cl^{2d( \frac{p−1}{p})}(ρ^{Nϵ}(u))^{\frac{l^{d}}{2}} − mτ^{u}1_{A_{k}}

+ ∑_{i=1}^{d} F β_{i}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(\frac{ρ_{β}^{k,l}(u + Nϵe_{i}) − ρ_{β}^{k,l}(u − Nϵe_{i})}{2Nϵ + 1})

+ F ω_{i}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(\frac{ρ_{ω}^{k,l}(u + Nϵe_{i}) − ρ_{ω}^{k,l}(u − Nϵe_{i})}{2Nϵ + 1})

G_{N,ϵ}^{4} = ∫_{0}^{T} \frac{1}{N^{d}} ∑_{u} −(Q_{g}(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u), ϕ(\frac{u}{N}), ψ(\frac{u}{N}))

− ((ϕ(\frac{u}{N}), ψ(\frac{u}{N})), A(ρ_{β}^{Nϵ}(u), ρ_{ω}^{Nϵ}(u))(ϕ(\frac{u}{N}), ψ(\frac{u}{N})))dt,

where ρ_{β}^{k,l} = \frac{1}{(2k−1)^{d}} ∑_{y∈B_{k−1}} β(y)τ^{y}1_{A_{l}} and ρ_{ω}^{k,l} is the natural analogue. Notice that the summations in G^{1} and G^{3} have been rearranged. Using Itô’s formula we can show that

\overline{lim}_{N→∞} E^{Q_{N}} [G_{N}^{2}] = 0.

By (0.14),

\overline{lim}_{l→∞} \overline{lim}_{m→∞} \overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} E^{Q_{N}} [G_{N,ϵ,k,m,l}^{3}] = 0.

By Theorem 0.4,

\overline{lim_{M→∞}} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} E^{Q_{N}} [max_{h≤M} G_{N,ϵ}^{4}] = 0.

Therefore we have that

\overline{lim}_{k→∞} \overline{lim}_{ϵ→0} \overline{lim}_{N→∞} E^{Q_{N}} [max_{i,j≤M} G_{N,ϵ,k}^{1}] ≤ log 3.

As functionals G_{N,ϵ,k}^{1} converge uniformly as N → ∞ to

G_{ϵ,k}^{1} =< (ϕ, γβ(T)) > + < (ψ, γω(T)) >

− ∫_{T^{d}} ∫_{0}^{T} ∑_{i=1}^{d} F β_{i}(\frac{μβ(θ − ϵ, θ + ϵ)}{2ϵ}, \frac{μω(θ − ϵ, θ + ϵ}{2ϵ})(\frac{ρ_{β}(θ + ϵe_{i}) − ρ_{β}(θ − ϵe_{i})}{2ϵ})

+ F ω_{i}(\frac{μβ(θ − ϵ, θ + ϵ)}{2ϵ}, \frac{μω(θ − ϵ, θ + ϵ)}{2ϵ})(\frac{ρ_{ω}(θ + ϵe_{i}) − ρ_{ω}(θ − ϵe_{i})}{2ϵ}) + o(k)

and we have

\overline{lim}_{k→∞} \overline{lim}_{ϵ→0} EQ̃[max_{i,j≤M} G_{ϵ,k}^{1}] ≤ log 3.

Using Lemma 14 we apply the dominated convergence theorem to take the limits inside the expectation. Finally, letting M → ∞ we get

EQ̃[sup_{ϕ,ψ} < (ϕ, γβ(T)) > + < (ψ, γω(T)) >

− \frac{1}{2} ∫_{T^{d}} ∫_{0}^{T} ((h′′A)(ρ_{β}(θ), ρ_{ω}(θ)) ( \begin{matrix}ϕ(θ) \\ ψ(θ) \end{matrix} ) , ( \begin{matrix}∇ρ_{β}(θ) \\ ∇ρ_{ω}(θ) \end{matrix} ))dtdθ] ≤ log 3.

Together with (0.53) this gives the desired result. □

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