getdoc884c. 330KB Jun 04 2011 12:04:38 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 10, pages 259–291.

Journal URL

http://www.math.washington.edu/~ejpecp/

Hydrodynamic limit of zero range processes among

random conductances on the supercritical percolation

cluster

Alessandra Faggionato

∗

Abstract

We consider i.i.d. random variables {ω(b) : b ∈ E_d} parameterized by the family of bonds in Zd, d >2. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval[0,c0]. Assuming the probabilitymof the event {ω(b)> 0} to be supercritical and denoting byC(ω)the unique infinite cluster associated to the bonds with posi-tive conductance, we study the zero range process onC(ω)withω(b)–proportional probability rate of jumps along bond b. For almost all realizations of the environment we prove that the hydrodynamic behavior of the zero range process is governed by the nonlinear heat equation

∂_tρ = m∇ ·(D∇φ(ρ/m)), where the matrixD and the function φ are ω–independent. As byproduct of the above result and the blocking effect of the finite clusters, we discuss the bulk behavior of the zero range process onZd_{with conductance fieldω. We do not require any} ellip-ticity condition.

Key words: disordered system, bond percolation, zero range process, hydrodynamic limit, ho-mogenization, stochastic domination.

AMS 2000 Subject Classification:Primary 60K35, 60J27, 82C44.

Submitted to EJP on August 19, 2009, final version accepted March 13, 2010.

∗_{Dipartimento di Matematica “G. Castelnuovo", Università “La Sapienza”. P.le Aldo Moro 2, 00185 Roma, Italy. e–mail:}

1

Introduction

Percolation provides a simple and, at the same time, very rich model of disordered medium [G],

[Ke]. The motion of a random walker on percolation clusters has been deeply investigated in Physics (see [BH] and reference therein) and also numerous rigorous results are now available. In the last years, for the supercritical percolation cluster it has been possible to prove the convergence of the diffusively rescaled random walk to the Brownian motion for almost all realizations of the percolation [SS], [BB], [MP], improving the annealed invariance principle obtained in [DFGW]. We address here our attention to interacting random walkers, moving on the supercritical Bernoulli bond percolation cluster with additional environmental disorder given by random conductances (for recent results on random walks among random conductances see[BP],[M],[F]and reference therein).

Particle interactions can be of different kind. An example is given by site exclusion, the hydrody-namic behavior of the resulting exclusion process has been studied in[F]. Another basic example, considered here, is the zero range interaction: particles lie on the sites of the infinite cluster without any constraint, while the probability rate of the jump of a particle from site x to a neighboring site

y is given by g(η(x))ω(x,y), whereg is a suitable function onN,η(x)is the number of particles at sitex andω(x,y)is the conductance of the bond {x,y}. We suppose that the concuctances are i.i.d. random variables taking value in[0,c0].

The above exclusion and zero range processes are non–gradient systems, since due to the disorder the algebraic local current cannot be written as spatial gradient of some local function. Neverthe-less, thanks to the independence of the conductances from any bond orientation, one can study the hydrodynamic behavior avoiding the heavy machinery of non–gradient particle systems [V],

[KL][Chapter VII]. Indeed, in the case of exclusion processes, due to the above symmetry of the conductance field the infinitesimal variation of the occupancy numberη(x)is a linear combination of occupancy numbers. This degree conservation strongly simplifies the analysis of the limiting be-havior of the random empirical measure with respect to genuinely non–gradient disordered models as in [Q1], [FM], [Q2], and can be reduced to an homogenization problem [F]. In the case of zero–range processes, this degree conservation is broken. Nevertheless, due to the symmetry of the conductance field, adapting the method of the corrected empirical measure [GJ1] to the present contest one can reduce the proof of the hydrodynamic limit to an homogenization problem plus the proof of the Replacement Lemma. The resulting diffusive hydrodynamic equation does not depend on the environment and keeps memory on the particle interaction.

The homogenization problem has been solved in [F] also for more general random conductance fields. The core of the problem here is the proof of the Replacement Lemma. This technical lemma compares the particle density on microscopic boxes with the particle density on macroscopic boxes and it is a key tool in order to go from the microscopic scale to the macroscopic one. This comparison is usually made by moving particles along macroscopic paths by microscopic steps and then summing the local variations at each step. The resulting method corresponds to the so called Moving Particle Lemma and becomes efficient if the chosen macroscopic paths allow a spread–out particle flux, without any concentration in some special bond. While for a.a. ω any two points x,y in a box

Λ_N of side N centered at the origin can be connected inside the infinite cluster by a path γx,y of

length at most O(N) [AP], it is very hard (maybe impossible) to exhibit such a family of paths

uniformly in b. Due to this obstacle, we will prove the Moving Particle Lemma not in its standard form, but in a weaker form, allowing anyway to complete the proof of the Replacement Lemma. We point out that in this step we use some technical results of[AP], where the chemical distance inside the supercritical Bernoulli bond percolation cluster is studied. It is only here that we need the hypothesis of i.i.d. conductances. Extending part of the results of [AP], one would get the hydrodynamic limit of zero range processes among random conductances on infinite clusters of more general conductance fields as in[F].

We comment another technical problem we had to handle with. The discussion in[GJ1]refers to the zero range process on a finite toroidal grid with conductances bounded above and below by some positive constants, and some steps cannot work here due to the presence of infinite particles. A particular care has to be devoted to the control of phenomena of particle concentration and slightly stronger homogenization results are required.

Finally, in the Appendix we discuss the bulk behavior of the zero range process on Zd with i.i.d. random conductances in [0,c₀], in the case of initial distributions with slowly varying parameter. Due to the blocking effect of the clusters with finite size, the bulk behavior is not described by a nonlinear heat equation.

We recall that the problem of density fluctuations for the zero range process on the supercritical Bernoulli bond percolation cluster with constant conductances has been studied in[GJ2]. Recently, the hydrodynamic limit of other interacting particle systems onZd, or fractal spaces, with random conductances has been proved (cf.[F1],[FJL],[JL],[LF],[Val]). We point out the pioneering paper

[Fr], where J. Fritz has proved the hydrodynamic behaviour of a one-dimensional Ginzburg-Landau model with conservation law in the presence of random conductances.

2

Models and results

2.1

The environment

The environmentω modeling the disordered medium is given by i.i.d. random variables ω(b) :

b∈E_d

, parameterized by the setE_d of non–oriented bonds inZd,d>2. ωandω(b)are thought of as the conductance field and the conductance at bond b, respectively. We callQthe law of the fieldωand we assume thatω(b)∈[0,c0]forQ–a.a.ω, for some fixed positive constantc0. Hence, without loss of generality, we can suppose that Q is a probability measure on the product space

Ω:= [0,c₀]E_d

. Moreover, in order to simplify the notation, we write ω(x,y) for the conductance

ω(b)if b={x,y}. Note thatω(x,y) =ω(y,x). Consider the random graphG(ω) = V(ω),E(ω)

with vertex setV(ω)and bond setE(ω)defined as

E(ω):=

b∈E_d : ω(b)>0 ,

V(ω):=

x∈Zd : x ∈bfor someb∈E(ω) .

Assuming the probability Q(ω(b) > 0) to be supercritical, the translation invariant Borel subset

Ω₀ ⊂ Ω given by the configurationsω for which the graph G(ω) has a unique infinite connected component (cluster)C(ω)⊂V(ω)hasQ–probability 1[G]. Below, we denote byE(ω)the bonds inE(ω)connecting points ofC(ω)and we will often understand the fact thatω∈Ω₀.

For later use, givenc>0 we define the random fieldωˆc= ωˆc(b) : b∈Ed

as

ˆ

ωc(b) =

(

1 ifω(b)>c,

0 otherwise . (2.1)

Forc=0 we simply setωˆ:= ˆω0.

2.2

The zero range process on the infinite cluster

C

(

ω

)

We fix a nondecreasing functiong:N→[0,∞)such thatg(0) =0, g(k)>0 for allk>0 and

g∗:=sup

k∈N

|g(k+1)−g(k)|<∞. (2.2)

Given a realization ω of the environment, we consider the zero range process ηt on the graph

G(ω) = C(ω),E(ω)

where a particle jumps from x to y with rate g(η(x))ω(x,y). This is the Markov process with pathsη(t)in the Skohorod space D [0,∞),NC(ω)

whose Markov generator

L acts on local functions as

Lf(η) =X e∈B

X

x∈C(ω):

x+e∈C(ω)

g(η(x))ω(x,x+e)€f(ηx,x+e)−f(η)Š, (2.3)

whereB={±e1,±e2, . . . ,±ed},e1, . . . ,ed being the canonical basis ofZd, and where in general

ηx,y(z) =







η(x)−1, ifz=x,

η(y) +1, ifz= y,

η(z), ifz6=x,y.

We recall that a function f is called local if f(η)depends only onη(x)for a finite number of sites

x. SinceC(ω)is infinite, the above process is well defined only for suitable initial distribution. As discussed in[A], the process is well defined when the initial distribution has support on configura-tionsηsuch thatkηk:=P_x∈C(ω)η(x)a(x)<∞, a(·)being a strictly positive real valued function onC(ω)such that

X

x∈C(ω)

X

e∈B

ω(x,x+e)a(x+e)6M a(x)

for some positive constantM.

Givenϕ >0, set Z(ϕ) :=P_k>0ϕk/g(k)! where g(0)! = 1, g(k)!= g(1)g(2)· · ·g(k) fork >1. Since Z(ϕ) is an increasing function and g(k)! > g(1)k, there exists a critical valueϕc ∈(0,∞]

such thatZ(ϕ)<∞ifϕ < ϕcand Z(ϕ) =∞ifϕ > ϕc. Then, for 06ϕ < ϕc we define ¯νϕ as the product probability measure onNC(ω)such that

¯

ν_ϕ(η(x) =k) = 1

Z(ϕ)

ϕk

g(k)!, k∈N, x ∈ C(ω).

Taking for example α(x) = e−|x| in the definition of kηk one obtains that ¯ν_ϕ(kηk) < ∞, thus implying that the zero range process is well defined whenever the initial distribution is given by ¯ν_ϕ

or by a probability measureµ stochastically dominated by ¯ν_ϕ. In this last case, as proven in [A], by the monotonicity ofgone obtains that the zero range processηt starting fromµis stochastically

dominated by the zero range process ζ_t starting from ¯ν_ϕ, i.e. one can construct on an enlarged probability space both processesηt andζt s.t. ηt(x)6ζt(x) almost surely. Finally, we recall that

all measures ¯ν_ϕ are reversible for the zero range process and that ¯ν_ϕ(eθ η(x))<∞for someθ >0, thus implying that ¯ν_ϕ(η(x)k_{)<∞for allk>0 (cf. Section 2.3 of[KL]).}

As usually done, we assume that lim_ϕ↑ϕcZ(ϕ) = ∞. Then, cf. Section 2.3 in [KL], the function

R(ϕ):=ν¯ϕ(η(0))is strictly increasing and gives a bijection from[0,ϕc)to[0,∞). Givenρ∈[0,∞)

we will writeϕ(ρ)for the unique value such thatR(ϕ) =ρ. Then we set

νρ:=ν¯ϕ(ρ), φ(ρ):=νρ(g(ηx)) x∈ C(ω). (2.4)

As proven in Section 2.3 of[KL],φis Lipschitz with constantg∗.

2.3

The hydrodynamic limit

Given an integerN >1 and a probability measureµN onNC(ω), we denote byP_ω,µN the law of the zero range process with generatorN2L (see (2.3)) and with initial distributionµN (assuming this dynamics to be admissible). We denote byE_ω,µN the associated expectation. In order to state the hydrodynamic limit, we defineB(Ω)as the family of bounded Borel functions onΩand letDbe the

d×d symmetric matrix characterized by the variational formula

(a,Da) = 1

mψ∈infB(Ω)



 

X

e∈B∗ Z

Ω

ω(0,e)(a_e+ψ(τeω)−ψ(ω))2I0,e∈C(ω)Q(dω)



 

, ∀a∈Rd, (2.5)

whereB_∗denotes the canonical basis ofZd,

m:=Q(0∈ C(ω)) (2.6)

and the translated environmentτeωis defined asτeω(x,y) =ω(x+e,y+e)for all bonds{x,y}

inE_d. In general,I_Adenotes the characteristic function ofA.

The above matrixDis the diffusion matrix of the random walk among random conductances on the supercritical percolation cluster and it equals the identity matrix multiplied by a positive constant (see the discussion in[F]and references therein).

Theorem 2.1. For Q–almost all environments ω the following holds. Let ρ0 : Rd → [0,∞) be a

bounded Borel function and let{µN}N>1be a sequence of probability measures onNC(ω)such that for

allδ >0and all continuous functions G onRd with compact support (shortly G∈Cc(Rd)), it holds

lim

N↑∞µ

N N

−d X x∈C(ω)

G(x/N)η(x)−

Z

Rd

G(x)ρ0(x)d x > δ

=0 . (2.7)

Moreover, suppose that there existρ0,ρ∗,C0 >0such thatµN is stochastically dominated by νρ0 and

the entropy H(µN_|ν

ρ∗)is bounded by C0N

Then, for all t>0, G∈C_c(Rd)andδ >0, it holds

lim

N↑∞Pω,µ N

N

−d X x∈C(ω)

G(x/N)η_t(x)−

Z

Rd

G(x)ρ(x,t)d x

> δ

=0 , (2.8)

whereρ:Rd×[0,∞)→Ris assumed to be the unique weak solution of the heat equation

∂tρ=m∇ ·(D∇φ(ρ/m)) (2.9)

with boundary conditionρ0at t=0.

We define the empirical measureπN_{(η)associated to the particle configurationηas}

πN(η):= 1

Nd

X

x∈C(ω)

η(x)δx/N ∈ M(Rd),

whereM(Rd)denotes the Polish space of non–negative Radon measures onRd endowed with the vague topology (namely,νn→ν inM if and only ifνn(f)→ν(f)for each f ∈Cc(Rd)). We refer to

the Appendix of[S]for a detailed discussion about the spaceM endowed of the vague topology. We writeπN_t for the empirical measureπN(ηt), ηt being the zero range process with generatorN2L.

Then condition (2.7) simply means that underµN _{the random measureπ}N _{converges in probability}

toρ0(x)d x, while underPω,µN the random measureπN

t converges in probability toρ(x,t)d x, for

each fixedt >0. In order to prove the conclusion of Theorem 2.1 one only needs to show that the law of the random pathπN

· ∈D([0,T],M)weakly converges to the delta distribution concentrated on the path [0,T]∋ t →ρ(x,t)d x ∈ M (see[KL][Chapter 5]). It is this stronger result that we prove here.

Let us give some comments on our assumptions. We have restricted to increasing functions g in order to assure attractiveness and therefore that the dynamics is well defined whenever the initial distributions are stochastically dominated by some invariant measureν_ρ0. This simplifies also some technical estimates. One could remove the monotone assumption ong and choose other conditions assuring a well defined dynamics and some basic technical estimates involved in the proof, which would be similar to the ones appearing in[KL][Chapter 5].

The entropy bound H(µN|ν_ρ∗) 6 C₀Nd is rather restrictive. Indeed, given a locally Riemann in-tegrable bounded profile ρ0 :Rd →[0,∞), letµN be the product measure on NC(ω) with slowly varying parameter associated to the profileρ0/mat scaleN. Namely,µN is the product measure on

NC(ω)such that

µN(η(x) =k) =νρ0(x/N)/m(η(x) =k).

Due to the ergodicity ofQcondition (2.7) is fulfilled and, settingρ′:=sup_xρ0(x),µN is stochasti-cally dominated byν_ρ′_/m. On the other hand, the entropyH(µN|ν_ρ

∗)is given by 1

NdH(µ

N

|νρ∗) = 1

Nd

X

x∈C(ω)

ρ0(x/N)

m

h

logϕ ρ0(x/N)

m

−logϕ ρ∗ io

+

1

Nd

X

x∈C(ω)

n

logZ ϕ ρ_∗

)−logZ ϕ ρ0(x/N)

m

Hence, H(µN|µ_ρ∗) 6C₀Nd only if ρ0 approaches sufficiently fast the constantmρ∗ at infinity. All these technical problems are due to the infinite space. In order to weaken the entropic assumption one should proceed as in[LM]. Since here we want to concentrate on the Moving Particle Lemma, which is the real new problem, we keep our assumptions.

Finally, we have assumed uniqueness of the solution of differential equation (2.9) with initial con-ditionρ0. Results on uniqueness can be found in[BC],[KL][Chapter 5]and[Va]. Proceeding as in

[KL][Section 5.7]and using the ideas developed below, one can prove that the limit points of the sequence{πN

t}t∈[0,T]are concentrated on pathst →ρ(x,t)d x satisfying an energy estimate.

3

Tightness of

{

π

N_t

}

_t∈[0,T]

As already mentioned, in order to reduce the proof of Theorem 2.1 to the Replacement Lemma one has to adapt the method of the corrected empirical measure developed in[GJ1]and after that invoke some homogenization properties proved in[F]. The discussion in[GJ1]refers to the zero range process on a finite toroidal grid and has to be modified in order to solve technical problems due to the presence of infinite particles.

GivenN∈N₊, we defineL_N as the generator of the random walk among random conductancesω

on the supercritical percolation cluster, after diffusive rescaling. More precisely, we defineCN(ω) =

{x/N : x∈ C(ω)}and set

L_Nf(x/N) =N2 X

e∈B:

x+e∈C(ω)

ω(x,x+e)nf((x+e)/N)−f(x/N)o (3.1)

for all x ∈ C(ω) and f : CN(ω) →R. We denote by νωN the uniform measure onCN(ω) given

byνN

ω =N−d P

x∈CN(ω)δx. Below we will think of the operatorLN as acting on L

2_(νN

ω). We write

(·,·)_νN

ω andk · kL2(νωN)for the scalar product and the norm inL 2_(νN

ω), respectively. Note thatLN is a

symmetric operator, such that(f,−L_Nf)_νN

ω >0 for each nonzero function f ∈L 2_(νN

ω). In particular,

λI−L_N is invertible for eachλ >0. Moreover, it holds

(f,−L_Ng)_νN ω =

1 2

X

x∈C(ω)

X

e∈B:

x+e∈C(ω)

ω(x,x+e)

f(x+e)−f(x)

·

g(x+e)−g(x)

for all functions f,g∈L2(νN

ω). Givenλ >0,G ∈Cc∞(Rd)andN∈N+we defineGλN as the unique

element ofL2(ν_ωN)such that

λGλ_N−L_NGλ_N=Gλ, (3.2)

whereGλ is defined as the restriction toCN(ω)of the functionλG− ∇ · D∇G∈Cc∞(R d_).

Let us collect some useful facts on the functionGλ_N:

Lemma 3.1. Fixλ >0. Then, for each G∈C_c∞(Rd)it holds

(G_Nλ,−L_NG_Nλ)_νN

ω 6c(G,λ), (3.3)

kGλ_NkL1_(νN ω),kG

λ

NkL2_(νN

ω)6c(λ,G), (3.4)

kL_NGλ_NkL1_(νN

ω),kLNG λ

NkL2_(νN

for a suitable positive constant c(λ,G)depending on λand G, but not from N . Moreover, for Q–a.s. conductance fieldsωit holds

lim

N↑∞kG λ

N−GkL2_(νN

ω)=0 , ∀G∈C ∞

c (R

d_{), (3.6)}

lim

N↑∞kG λ

N−GkL1_(νN

ω)=0 , ∀G∈C ∞

c (R

d_{). (3.7)}

Proof. By taking the scalar product withG_Nλ in (3.2) one obtains that

λkG_Nλk2_L2_(νN ω)

+ (Gλ_N,−L_NGλ_N)_νN ω = (G

λ

N,G

λ₎ νN

ω 6kG λ

NkL2_(νN ω)kG

λ_k

L2_(νN ω). Using that sup_NkGλk_L2_(νN

ω) < ∞, from the above expression one easily obtains the uniform up-per bounds on(Gλ_N,−L_NGλ_N)_νN

ω andkG λ

NkL2_(νN

ω). Since supN>1kG λ_k

L2_(νN

ω)<∞, by difference one obtains the uniform upper bound onkL_NGλ_Nk_L2_(νN

ω). In order to estimatekG_NλkL1_(νN

ω) let us write p

N

t (x,y) for the probability that the random walk on

CN(ω) with generatorLN and starting point x is at site y at time t. Then, since the jump rates

depend on the unoriented bonds,pN_t (x,y) =pN_t (y,x). Since

G_Nλ(x) = X y∈CN(ω)

Z ∞

0

e−λtpN_t (x,y)Gλ(y), (3.8) for allx ∈ CN(ω), the above symmetry allows to conclude that

kGλ_NkL1_(νN ω)6

1

Nd

X

x,y∈CN(ω)

Z ∞

0

e−λtpN_t (x,y)|Gλ(x)|=

1

λNd

X

x∈CN(ω)

|Gλ(x)| → 1

λ

Z

Rd

|Gλ(u)|du<∞. (3.9)

Again, since sup_N>1kGλkL1_(νN

ω) < ∞, by difference one obtains the uniform upper bound on

kL_NGλ_NkL1_(νN ω).

The homogenization result (3.6) follows from Theorem 2.4 (iii) in[F]. Finally, let us consider (3.7). Givenℓ >0, using Schwarz inequality, one can bound

kGλ_N−Gk_L1_(νN ω)6

kG_Nλ(u)I(|u|> ℓ)kL1_(νN

ω)+kG(u)I(|u|> ℓ)kL1(νωN)+cℓ

d/2_kGλ

N−Gk

1/2

L2_(νN ω)

. SinceG∈C_c∞(Rd)the second term in the r.h.s. is zero forℓlarge enough. The last term in the r.h.s. goes to zero due to (3.6). In order to conclude we need to show that

lim ℓ↑∞Nlim↑∞kG

λ

N(u)I(|u|> ℓ)kL1_(νN

ω)=0 . (3.10)

SinceGλ ∈C_c∞(Rd) we can fix nonnegative functionsF,f ∈C_c∞(Rd) such that−f 6Gλ6 F. We callF_Nλ, f_Nλ the solutions in L2(ν_ωN)of the equations

λF_Nλ−L_NF_Nλ=F,

respectively. From the integral representation (3.8) we derive thatF_Nλ, f_Nλ are nonnegative, and that

−f_Nλ 6Gλ_N 6F_Nλ on CN(ω). In particular, in order to prove (3.10) it is enough to prove the same

claim withF_Nλ, f_Nλ instead ofGλ_N. We give the proof forF_Nλ, the other case is completely similar. Let us defineH ∈C∞(Rd)as the unique solution in L2(d x)of the equation

λH− ∇ · D∇H=F. (3.11)

Again, by a suitable integral representation, we get that H is nonnegative. Applying Schwarz in-equality, we can estimate

kF_Nλ(u)I(|u|> ℓ)kL1_(νN ω)=kF

λ

NkL1_(νN ω)− kF

λ

N(u)I(|u|6ℓ)kL1_(νN ω)6

kF_NλkL1_(νN

ω)− kH(u)I(|u|6ℓ)kL1(νωN)+k(H(u)−F λ

N)I(|u|6ℓ)kL1_(νN ω)6

kF_NλkL1_(νN

ω)− kH(u)I(|u|6ℓ)kL1(νωN)+cℓ

d/2

kF_Nλ−Hk1_L/22_(νN ω)

. (3.12) SinceF_Nλ andF are nonnegative functions, when repeating the steps in (3.9) withF_Nλ, F instead of

Gλ_N,Gλ respectively, we get the the inequality is an equality and therefore

kF_NλkL1_(νN ω)→λ

−1

kFkL1_{(d x)}=kHk_L1_{(d x)}.

This observation, the above bound (3.12) and Theorem 2.4 (iii) in[F]imply that lim

N↑∞kF λ

N(u)I(|u|> ℓ)kL1_(νN

ω)6kH(u)I(|u|> ℓ)kL1(νωN) At this point it is trivial to derive (3.10) forF_Nλ.

In the rest of this section, we will assume thatωis a good conductance field, i.e. the infinite cluster

C(ω)is well–defined andωsatisfies Lemma 3.1. We recall that these properties holdQ–a.s. The first step in proving the hydrodynamic limit consists in showing that the sequence of processes

{πN_t}t∈[0,T]is tight in the Skohorod spaceD([0,T],M). By adapting the proof of Proposition IV.1.7 in[KL]to the vague convergence, one obtains that it is enough to show that the sequence of pro-cesses{πN_t[G]}t∈[0,T]is tight in the Skohorod space D([0,T],R)for allG∈Cc∞(R). A key relation

between the zero range process and the random walk among random conductances is given by

N2L€πN(η)[G]Š= 1

Nd

X

x∈C(ω)

g(η(x)) L_NG

(x/N). (3.13)

The check of (3.13) is trivial and based on integration by parts. At this point, due to the disorder given by the conductance fieldω, a second integration by parts as usually done for gradient systems (cf. [KL][Chapter 5]) would be useless since the resulting object would remain wild. A way to over-come this technical problem is given by the method of the corrected empirical measure: as explained below, the sequence of processes {πN_t [G]}t∈[0,T] behaves asymptotically as {πNt [GλN]}t∈[0,T], thus the tightness of the former follows from the tightness of the latter. We need some care since the total number of particles can be infinite, hence it is not trivial that the process{πN_t[G_Nλ]}t∈[0,T] is well defined.

Lemma 3.2. Let H be a nonnegative function onCN(ω)belonging to L1(νωN)∩L2(νωN)and let k>0.

Then

P_ω,µN

sup 06t6T

1

Nd

X

x∈C(ω)

ηt(x)kH(x/N)>A

6A−1c(k,ρ0) q

kHk2_L1_(νN ω)

+kHk2_L2_(νN ω)

(3.14)

for all A>0.

Proof. We use a maximal inequality for reversible Markov processes due to Kipnis and Varadhan

[KV](cf. Theorem 11.1 in Appendix 1 of[KL]). Let us set

F(η) = 1

Nd

X

x∈C(ω)

η(x)kH(x/N), (3.15)

supposing first that H has bounded support. Note that F(η) 6 F(η′) if η(x) 6 η′(x) for all

x ∈ C(ω). Hence by the stochastic domination assumption, it is enough to prove (3.14) with

P_ω,ν

ρ0 (always referred to the diffusively accelerated process) instead ofPω,µ

N. We recall thatν_ρ 0 is reversible w.r.t. the the zero range process. Moreover

ν_ρ0(F2) = 1

N2d

X

x,y∈C(ω)

H(x/N)H(y/N)ν_ρ0(η(x)kη(y)k)6c(k,ρ0)kHk2_L1_(νN ω)

, (3.16)

while

νρ0(F,−N

2

LF) = N

2

N2d

X

x,y∈C(ω)

H(x/N)H(y/N)νρ0(η(x)

k_,−Lη(y)k₎₆

c(k,ρ0)

N2 N2d

X

x∈C(ω)

X

y∈C(ω): |x−y|=1

H(x/N)H(y/N).

Using the boundH(x/N)H(y/N)6H(x/N)2+H(y/N)2and the fact thatd>2, we conclude that

ν_ρ0(F,−N2LF)6c(k,ρ0)kHk2_L2_(νN ω)

. (3.17)

By the result of Kipnis and Varadhan it holds

P_ω,ν

ρ0

sup 06t6T

1

Nd

X

x∈C(ω)

ηt(x)kH(x/N)>A

6 e

A

Æ

ν_ρ0(F2) +Tν_ρ0(F,−N2_{LF). (3.18)}

At this point the thesis follows from the above bounds (3.16) and (3.17). In order to remove the assumption that H is local, it is enough to apply the result to the sequence of functionsH_n(x) :=

H(x)χ(|x|6n)and then apply the Monotone Convergence Theorem asn↑ ∞.

Remark 1. We observe that the arguments used in the proof of Lemma 4.3 in[CLO]imply that, given

a function H of bounded support and defining F as in(3.15), it holds

E_ω,ν

ρ0

sup 06t6T

F(ηt)−F(η0) 2

In particular, it holds

E_ω,ν

ρ0

sup 06t6T

F(ηt)2

6cν_ρ0(F2) +c Tν_ρ0(F,−N2LF).

Using the bounds(3.16)and(3.17), the domination assumption and the Monotone Convergence The-orem, under the same assumption of Lemma 3.2 one obtains

E_ω,µN

sup 06t6T

h 1

Nd

X

x∈C(ω)

ηt(x)kH(x/N)

i2

6c(k,ρ0) h

kHk2_L1_(νN ω)

+kHk2_L2_(νN ω)

i

. (3.19)

Using afterwards the Markov inequality, one concludes that

P_ω,µN

sup 06t6T

1

Nd

X

x∈C(ω)

ηt(x)kH(x/N)>A

6c(k,ρ0)A−2 h

kHk2_L1_(νN ω)

+kHk2_L2_(νN ω)

i

(3.20)

for all A>0. The use of (3.14)or(3.20)in the rest of the discussion is completely equivalent.

Due to Lemma 3.2 and Lemma 3.1 the process{πN_t[G_Nλ]}t∈[0,T] is well defined w.r.t Eω,µN. Let us explain why this process is a good approximation of the process{πN_t[G]}t∈[0,T]:

Lemma 3.3. Let G∈C_c∞(Rd). Then, givenδ >0, it holds

lim

N↑∞Pω,µ N

sup 06t6T

πN_t [Gλ_N]−πN_t[G] > δ

=0 . (3.21)

Proof. By Lemma 3.2 we can bound the above probability by

c(ρ0)δ−1 q

kGλ

N−Gk2L1_(νN ω)

+kGλ

N−Gk2L2_(νN ω)

. At this point the thesis follows from Lemma 3.1.

Due to the above Lemma, in order to prove the tightness of{πN_t[G]}t∈[0,T]it is enough to prove the tightness of{πN_t [Gλ_N]}t∈[0,T]. Now we can go on with the standard method based on martingales and Aldous criterion for tightness (cf. [KL][Chapter 5]), but again we need to handle with care our objects due to the risk of explosion. We fix a good realizationωof the conductance field. Due to Lemma 3.1, Lemma 3.2 and the bound g(k)6 g∗k, we conclude that the process{M_tN}06t6T

where

M_tN(G):=πN_t (Gλ_N)−π₀N(Gλ_N)−

Z t

0 1

Nd

X

x∈C(ω)

g(η_s(x))L_NG_Nλ(x/N)ds, (3.22)

is well definedP_ω,µN–a.s.

Lemma 3.4. Givenδ >0,

lim

N↑∞Pω,µ N

sup 06t6T

|M_tN(G)|>δ

Proof. Givenn>1, we define the cut–off functionG_Nλ_,n:CN(ω)→RasGNλ,n(x) =GNλ(x)I(|x|6n).

Then G_Nλ_,n is a local function and by the results of [A] (together with the stochastic domination assumption) we know that

M_tN,n(G):=πN_t(G_Nλ_,n)−πN₀(G_Nλ_,n)−

Z t

0 1

Nd

X

x∈C(ω)

g(ηs(x))LNGλN,n(x/N)ds

is anL2–martingale of quadratic variation

<M_tN,n(G)>=

Z t

0

N2 N2d

X

x∈C(ω)

X

y∈C(ω): |x−y|=1

g(η_s(x))ω(x,y)

Gλ_N,n(y/N)−Gλ_N,n(x/N)2ds.

Note that, by the stochastic domination assumption and the bound g(k)6g∗k,

E_ω,µN <M_tN,n(G)>6

g∗

Z t

0

N2 N2d

X

x∈C(ω)

X

y∈C(ω): |x−y|=1

E_ω,νρ

0

ηs(x)

ω(x,y)

Gλ_N,n(y/N)−G_Nλ_,n(x/N)2ds

=g∗ρ0t N−d(GλN,n,−LNGNλ,n)νN

ω. (3.24) By Doob’s inequality and (3.24), we conclude that

P_ω,µN

sup 06t6T

|MtN,n(G)|>δ

6 c

δ2Eω,µN |M

N,n T (G)|

2

6

c g∗Tρ0

δ2_Nd (G

λ

N,n,−LNGλN,n)νN ω 6

c′g∗Tρ0

δ2_Nd . (3.25)

Above we have used that lim_n↑∞(Gλ_N,n,−L_NGλ_N,n)_νN ω = (G

λ

N,−LNGNλ)νN

ω 6c(λ)(see Lemma 3.1). The above process{M_tN,n(G)}t∈[0,T]is a good approximation of{MtN(G)}t∈[0,T]asn↑ ∞. Indeed, it holds

lim

n↑∞Pω,µ N

sup 06t6T

|M_tN,n(G)−M_tN(G)|> δ=0 , δ >0 . (3.26)

Indeed, sincekG_Nλ_,n−G_Nλk_L1_(νN

ω),L2(νωN)andkLNG λ

N,n−LNGNλkL1_(νN

ω),L2(νωN)converge to zero asN↑ ∞ and since g(k)6g∗k, the above claim follows from Lemma 3.2.

At this point, (3.26) and (3.25) imply (3.23).

Let us prove the tightness of{πN_t[Gλ_N]}t∈[0,T]using Aldous criterion (cf. Proposition 1.2 and Propo-sition 1.6 in Section 4 of[KL]):

Proof. Fixθ >0 and uppose thatτis a stopping time w.r.t. the canonical filtration bounded byT. With some abuse of notation we writeτ+θ for the quantity min{τ+θ,T}. Then, givenǫ >0, by Lemmata 3.1 and 3.2,

P_ω,µN

Z τ+θ

τ 1

Nd

X

x∈C(ω)

g(ηs(x))LNGNλ(x/N)ds

> ǫ

6

P_ω,µN θg∗ sup

s∈[0,T] 1

Nd

X

x∈C(ω)

ηs(x)|LNGNλ(x/N)|ds> ǫ

6C g∗θ /ǫ. (3.27)

In particular, lim

γ↓0lim sup_{N↑∞ τ,θ}sup_∈[0,γ]Pω,µ

N

Z τ+θ

τ 1

Nd

X

x∈C(ω)

g(ηs(x))LNGλN(x/N)ds

> ǫ

=0 . (3.28) An estimate similar to (3.27) implies that

lim ǫ↑∞Nsup>1

P_ω,µN

Z t 0 1 Nd X

x∈C(ω)

g(ηs(x))LNGλN(x/N)ds

> ǫ

=0 . (3.29)

Let us now come back to Lemma 3.4. Letτ,θ as above. Then lim sup

N↑∞

P_ω,µN |MN τ+θ−M

N

τ|> ǫ

6lim sup

N↑∞

P_ω,µN sup

06s6T

|M_sN|> ǫ/2=0 (3.30)

Collecting (3.28), (3.29) and (3.30), together with Lemma 3.4, we conclude that lim

γ↓0lim sup_{N↑∞ τ,θ∈}sup_[0,γ]Pω,µ

N

πN

τ+θ[GNλ]−π N

τ[GNλ]

> ǫ

=0 , (3.31)

lim ǫ↑∞Nsup>1

P_ω,µN |πN_t[G_Nλ]> ǫ

=0 . (3.32)

Aldous criterion for tightness allows to derive the thesis from (3.31) and (3.32).

Let us come back to (3.22) and investigate the integral term there. The following holds:

Lemma 3.6. Let I(t):=Rt

0N −dP

x∈C(ω)g(ηs(x)) GNλ(x/N)−G(x/N)

ds. Then, for allδ >0,

lim

N↑∞Pω,µ

N sup

06t6T

|I(t)|> δ

=0 . (3.33)

Proof. Sinceg(k)6g∗kand by Schwarz inequality we can bound

I(t)6J:=T g∗kG_Nλ−Gk1_L/12_(νN ω)

sup 06s6T

{N−d X

x∈C(ω)

η_s(x)2G_Nλ(x/N)−G(x/N) }1/2. Using the stochastic domination assumption and applying Lemma 3.2 we obtain

P_ω,µN sup

06t6T

|I(t)|> δ

6P_ω,ν

ρ0(J > δ)6 (1/δ)2T2(g∗)2kG_Nλ−GkL1_(νN

ω) q

kG_Nλ−Gk2_L1_(νN ω)

+kGλ_N−Gk2_L2_(νN ω)

. (3.34) The thesis now follows by applying Lemma 3.1.

We are finally arrived at the conclusion. Indeed, due to Lemma 3.3 and Lemma 3.5 we know that the sequence of processes{πN_t}t∈[0,T]is tight in the Skohorod space D([0,T],M). Moreover, starting from the identity (3.22), applying Lemma 3.4, using the identity (3.2) which equivalent to

L_NG_Nλ =λG_Nλ−Gλ=λ(G_Nλ−G) +∇ · D∇G,

and finally invoking Lemma 3.6 we conclude that, fixed a good conductance field ω, for anyG ∈ C_c∞(Rd)and for anyδ >0

lim

N↑∞Pω,µN

sup 06t6T

π

N

t(G)−π N

0(G)− Z t

0 1

Nd

X

x∈C(ω)

g(ηs(x))∇ · D∇G(x/N)ds

> δ

=0 . (3.35)

Using the stochastic domination assumption it is trivial to prove that any limit point of the sequence

{πN

t}t∈[0,T] is concentrate on trajectories {πt}t∈[0,T] such that πt is absolutely continuous w.r.t.

to the Lebesgue measure. Moreover, in order to characterize the limit points as solution of the differential equation (2.9) one would need non only (3.35). Indeed, it is necessary to prove that, givenωgood, for each functionG∈C_c∞ [0,T]×Rd

it holds lim

N↑∞Pω,µ N

sup 06t6T

π

N

t (Gt)−πN0(G0)− Z t

0 1

Nd

X

x∈C(ω)

g(ηs(x))∇ · D∇Gs(x/N)ds−

Z t

0

πN_s (∂sGs)ds

> δ

=0 , (3.36)

where Gs(x):=G(s,x). One can easily recover (3.36) from the same estimates used to get (3.35)

and suitable approximations ofG which are piecewise linear int as in the final part of Section 3 in

[GJ1]. In order to avoid heavy notation will continue the investigation of (3.35) only.

4

The Replacement Lemma

As consequence of the discussion in the previous section, in order to prove the hydrodynamical limit stated in Theorem 2.1 we only need to control the term

Z t

0 1

Nd

X

x∈C(ω)

g(η_s(x))∇ · D∇G(x/N)ds. (4.1)

To this aim we first introduce some notation. Given a family of parametersα1,α2, . . . ,αn, we will

write

lim sup α1→a1,α2→a2,...,αn→an instead of

lim sup αn→an

lim sup αn−1→an−1

· · ·lim sup α1→a1

. Below, given x∈Zd andk∈N, we writeΛ_x,kfor the box

and we writeηk(x)for the density

ηk(x):= 1 (2k+1)d

X

y∈Λx,k∩C(ω)

η(y).

If x=0 we simply writeΛ_kinstead ofΛ_0,k.

Then, we claim that for Q–a.a. ω, given G ∈ C_c(Rd), δ > 0 and a sequence µN of probability measures onNC(ω)stochastically dominated by someν_ρ0and such thatH(µN_|ν

ρ∗)6C0N

d_{, it holds}

lim sup

N↑∞,ǫ↓0

P_ω,µN

Z t

0 1

Nd

X

x∈C(ω)

g ηs(x)

G(x/N)ds−

Z t

0

m Nd

X

x∈Zd

φ ηǫ_sN(x)/mG(x/N)ds

> δ

=0 . (4.2)

Let us first assume the above claim and explain how to conclude, supposing for simplicity of notation thatǫN∈N.

Givenu∈Rd andǫ >0, defineιu,ǫ:= (2ǫ)−dI{u∈[−ǫ,ǫ]d}. Then the integralπN

ιx/N,ǫ

,x ∈Zd, can be written as

πN

ιx/N,ǫ

=(2ǫN+1) d (2ǫN)d η

ǫN_{(x). (4.3)}

Let us define

Z t

0

m Nd

X

x∈Zd

φ(πN_s [ιx/N,ǫ/m])∇ · D∇G(x/N)ds. (4.4)

Then, due to (4.3) and since φ is Lipschitz with constant g∗, we can estimate from above the difference between (4.4) and the second integral term in (4.2) withGsubstituted by∇ · D∇Gas

(2ǫN+1)d_−(2ǫN)d (2ǫN)d

Z t

0 1

Nd

X

x

ηǫ_sN(x)∇ · D∇G(x/N)

ds (4.5)

Since the integral term in (4.5) has finite expectation w.r.t P_ω,ν

ρ0 and therefore also w.r.t. Pω,µ N, we conclude that the above difference goes to zero in probability w.r.t. P_ω,µN. At this point the conclusion of the proof of Theorem 2.1 can be obtained by the same arguments used in[KL][pages 78,79].

Let us come back to our claim. Since X

x∈C(ω)

g(ηs(x))G(x/N) =

X

x∈Zd

g(ηs(x))I(x∈ C(ω))G(x/N),

by a standard integration by parts argument and using thatG ∈C_c∞(Rd)one can replace the first integral in (4.2) by

Z t

0 1

Nd

X

x∈Zd

h 1

(2ℓ+1)d

X

y:y∈Λx,ℓ∩C(ω)

g ηs(y)

i

G(x/N).

Lemma 4.1. (Replacement Lemma) ForQ–a.a. ω, givenδ >0, M∈Nand given a sequence of prob-ability measuresµNonNC(ω)stochastically dominated by someν_ρ0and such that H(µN|ν_ρ∗)6C0Nd,

it holds

lim sup

N↑∞,ǫ↓0

P_ω,µN

hZ t

0 1

Nd

X

x∈ΛM N

V_ǫN(τxηs,τxω)ds> δ

i

=0 , (4.6)

where

V_ℓ(η,ω) =

1

(2ℓ+1)d

X

y:y∈Λℓ∩C(ω)

g η(y)

−mφ ηℓ(0)/m .

Let us defineΥ_C

0,N as the set of measurable functions f :N

C(ω)_{→[0,∞)such that i)ν}

ρ∗(f) =1, (ii)D(f):=ν_ρ∗(pf,−Lpf)6C0Nd−2and (iii) f dνρ∗ is stochastically dominated byνρ0(shortly,

f dνρ∗ ≺dνρ0). Using the assumption H(µ

N_|ν

ρ∗)6C0N

d _{and entropy production arguments as in}

[KL][Chapter 5], in order to prove the Replacement Lemma it is enough to show that forQ–a.a.ω, givenρ0,ρ∗,C0,M>0, it holds

lim sup

N↑∞,ǫ↓0 sup

f∈ΥC0,N

Z 1

Nd

X

x∈ΛM N

V_ǫN(τxη,τxω)f(η)νρ∗(dη) =0 . (4.7)

Trivially, since ν_ρ1 stochastically dominates ν_ρ2 if ρ1 > ρ2, it is enough to prove that, given

ρ0,ρ∗,C0,M > 0, for Q–a.a. ω (4.7) is verified. We claim that the above result follows from the the One Block and the Two Blocks estimates:

Lemma 4.2. (One block estimate) Fixρ0,ρ∗,C0,M >0. Then, forQ–a.a.ωit holds

lim sup

N↑∞,ℓ↑∞ sup

f∈ΥC0,N

Z 1

Nd

X

x∈ΛM N

V_ℓ(τxη,τxω)f(η)νρ∗(dη) =0 . (4.8)

Lemma 4.3. (Two blocks estimate)Fixρ0,ρ∗,C0,M>0. Then, forQ–a.a. ωit holds

lim sup

N↑∞,ǫ↓0,ℓ↑∞ sup

f∈ΥC_0,N

Z 1

Nd

X

x∈ΛM N

h 1

(2ǫN+1)d

X

y∈Λx,ǫN

ηℓ(y)−ηǫN(x) i

f(η)ν_ρ∗(dη) =0 . (4.9)

We point out that the form of the Two Blocks Estimate is slightly weaker from the one in

[KL][Chapter 5], nevertheless it is strong enough to imply, together with the One Block Estimate, equation (4.7). Indeed, let us definea(y):=I(y ∈ C(ω))and

I1(η):= 1

Nd

X

x∈ΛM N

Avy∈Λx,ǫNg(η(y))a(y)−Avy∈Λx,ǫNAvz∈Λy,ℓg(η(z))a(z)

, (4.10)

I2(η):= 1

Nd

X

x∈ΛM N

Avy∈Λx,ǫN

Avz∈Λy,ℓg(η(z))a(z)−mφ(η

ℓ_(y)/m)

, (4.11)

I₃(η):= 1

Nd

X

x∈ΛM N

Avy∈Λx,ǫNmφ(η

whereAvdenotes the standard average. Then 1

Nd

X

x∈ΛM N

V_ǫN(τxη,τxω) =

1

Nd

X

x∈ΛM N

Avy∈Λx,ǫNg(η(y))a(y)−mφ(η

ǫN_(x)/m)

6(I1+I2+I3)(η). (4.13)

Let us explain a simple bound that will be frequently used below, often without any mention. Con-sider a family of numbersb(x), x ∈Zd. Then, taking L,ℓ >0 we can write

Avx∈ΛLb(x)−Avx∈ΛLAvu∈Λx,ℓb(u) = 1

|Λ_L|

X

x∈ΛL+ℓ

b(x) I(x∈Λ_L)− 1

|Λ_ℓ|♯{u∈ΛL:|x−u|∞6ℓ}

.

In particular, it holds Av_x∈Λ

Lb(x)−Avx∈ΛLAvu∈Λx,ℓb(u) 6

1

|Λ_L|

X

x∈ΛL+ℓ\ΛL−ℓ

|b(x)|. (4.14)

Due to the above bound we conclude that

I1(η)6

g∗ Nd_(2ǫN+1)d

X

x∈ΛM N

X

u∈Λx,ǫN+ℓ\Λx,ǫN−ℓ

η(u)a(u).

In particular, using that f dν_ρ∗ ≺dν_ρ0, we conclude thatRI1(η)f(η)νρ∗(dη)6cℓ/(ǫN). The sec-ond termI2(η)can be estimated forǫ61 as

I2(η)6 1

Nd

X

x∈ΛM N

Avy∈Λx,ǫNVℓ(τzη,τzω)6 1

Nd

X

x∈Λ(M+1)N

V_ℓ(τxη,τxω).

Due to the One block estimate one gets that lim sup

N↑∞,ǫ↓0,ℓ↑∞ sup

f∈ΥC0,N

Z

I₂(η)f(η)ν_ρ∗(dη) =0 .

The same result holds also forI3(η)due to the Lipschitz property ofφand the Two Blocks estimate. The above observations together with (4.13) imply (4.7).

5

Proof of the Two Blocks Estimate

For simplicity of notation we setℓ_∗= (2ℓ+1)d and we take M=1 (the general case can be treated similarly). Moreover, given ∆⊂Zd, we writeN(∆)for the number of particles in the region∆, namelyN(∆):=P_{x∈∆∩C(ω)}η(x).

Let us set

A(η):= 1

Nd

X

x∈ΛN

Avy∈Λx,ǫN

ηℓ(y)−ηǫN(x)

, (5.1)

B(η):= 1

Nd

X

x∈ΛN

Avy∈Λx,ǫNAvz∈Λx,ǫN

ηℓ(y)−ηℓ(z)

, (5.2)

C(η):= 1

Nd_|Λ

ǫN|

X

x∈ΛN

Av_y∈Λ

x,ǫN

X

z∈Λx,ǫN: |z−y|∞>2ℓ

ηℓ(y)−ηℓ(z)

. (5.3)

Since due to (4.14)

ηǫN(x) =Av_z∈Λx,ǫNη

ℓ_{(z) +E, |E |6c|Λ} ǫN|−1

X

u∈Λx,ǫN+ℓ\Λx,ǫN−ℓ

η(u),

using that f dν_ρ∗≺dν_ρ0, in order to prove the Two Blocks estimate we only need to show that

lim sup

N↑∞,ǫ↓0,ℓ↑∞ sup

f∈ΥC_0,N

Z

B(η)f(η)ν_ρ∗(dη) =0 .

Since

B(η)6C(η) + 1

Nd_|Λ

ǫN|

X

x∈ΛN

Av_y∈Λ

x,ǫN

X

z∈Λx,ǫN: |z−y|∞62ℓ

ηℓ(y) +ηℓ(z) ,

using again that f dν_ρ∗ ≺ dν_ρ0, in order to prove the Two Blocks Estimate we only need to show that

lim sup

N↑∞,ǫ↓0,ℓ↑∞ sup

f∈ΥC0,N

Z

1

Nd_(ǫN)2d_ℓd

X

x∈ΛN

X

y∈Λx,ǫN

X

z∈Λx,ǫN: |z−y|∞>2ℓ

N(Λ_y,ℓ)− N(Λ_z,ℓ) f(η)ν_ρ

∗(dη) =0 . (5.4)

Let us now make an observation that will be frequently used below. LetX ⊂Zd be a subset possibly depending onωand on some parameters (for simplicity, we consider a real–value parameterL∈N). Suppose that forQ–a.a.ωit holds

lim sup

N↑∞,L↑∞

|X∩Λ_N| Nd =0 .

Then, in order to prove (5.4) we only need to show that lim sup

N↑∞,ǫ↓0,ℓ↑∞,L↑∞ sup

f∈ΥC0,N

Z

1

Nd_(ǫN)2d_ℓd

X

x∈ΛN

X

y∈Λx,ǫN

X

z∈Λx,ǫN: |z−y|∞>2ℓ

N(Λ_y,ℓ\X)− N(Λ_z,ℓ\X) f(η)ν_ρ

We know that there existsα0>0 such that for eachα∈(0,α0]the random fieldωˆαdefined in (2.1) is a supercritical Bernoulli bond percolation. Let us writeC_α(ω)for the associated infinite cluster. By ergodicity,

lim sup

N↑∞,α↓0

|(C \ C_α)∩Λ_N|/|Λ_N|=lim

α↓0Q(0∈ C \ Cα) =0 . (5.6)

Hence, due to the above considerations, (5.4) is proven if we show that, for each α ∈(0,α0], it holds (5.4) withC replaced byC_α. Moreover, since f dν_ρ∗ ≺ν_ρ0, using Chebyshev inequality it is simple to prove that

lim sup

N↑∞,ǫ↓0,ℓ↑∞,A↑∞ Z

1

Nd_(ǫN)2d

X

x∈ΛN

X

y∈Λx,ǫN

X

z∈Λx,ǫN: |z−y|∞>2ℓ

I(N(Λ_y,ℓ∪Λ_z,ℓ)>Aℓ_∗d)f(η)ν_ρ∗(dη) =0 . At this point, we only need to prove the following: Fixedα ∈(0,α0]and A>0, forQ–a.a. ω it holds

lim sup

N↑∞,ǫ↓0,ℓ↑∞ sup

f∈Υ∗ C_0,N

Z

1

Nd(ǫN)2d_ℓd

X

x∈ΛN

X

y∈Λx,ǫN

X

z∈Λx,ǫN: |z−y|∞>2ℓ

N(C_α∩Λ_y,ℓ)− N(C_α∩Λ_z,ℓ)

I(N(Λ_y,ℓ∪Λ_z,ℓ)6Aℓd

∗)f(η)νρ∗(dη) =0 , (5.7) where Υ∗_C

0,N is the family of measurable functions f :N

C(ω) _{→[0,∞) such that ν}

ρ∗(f) =1 and

D(f)6C0Nd−2.

We now use the results of [AP] about the chemical distance in the supercritical Bernoulli bond percolation ωˆ_α, for some fixed α ∈ (0,α0]. We fix a positive integer K (this corresponds to the parameterN in [AP], which is fixed large enough once for all). Given a∈Zd ands >0, we set

∆_a,s := Λ_(2K+1)a,s. As in [AP], we call ωˆ_α the microscopic random field. The macroscopic one

σ={σ(a):a∈Zd} ∈ {0, 1}Zd

is defined in[AP]stating thatσ(a) =1 if and only if the microscopic fieldωˆ_αsatisfies certain geometric properties inside the box∆_a,5K/4. These properties are described on page 1038 in [AP], but their content is not relevant here, hence we do not recall them. What is relevant for us is that there exists a function ¯p : N → [0, 1) with limK↑∞¯p(K) = 1, such that

σ stochastically dominates a Bernoulli site percolation of parameter ¯p(K) (see Proposition 2.1 in

[AP]). Below we denote byP_¯p(K)the law of this last random field, takingK large enough such that ¯

p(K)is supercritical. As in[AP]we call a pointa∈Zd white or black ifσ(a) =1 or 0 respectively, and we write in boldface the sites referred to the macroscopic field. Recall that a subset ofZd is

∗–connected if it is connected with respect to the adjacency relation

x ∼∗ y⇔ |x−y|∞=1 .

C∗is defined as the set of all∗–connected macroscopic black clusters. Givena∈Zd,C∗_adenotes the element ofC∗ containing a(with the convention that C∗_a =;ifais white), while ¯C∗_a is defined as ¯

C∗_a=C∗_a∪∂outC∗_a. We recall that, given a finite subsetΛ⊂Zd, its outer boundary is defined as

∂outΛ:=

Again by conditioning on the number of particles in Γ_x,ℓ,α and afterwards applying perturbation theory (see (5.29), (5.31) and (5.33)), one only needs to show that

lim sup N↑∞,ℓ↑∞,L↑∞

Av_x∈ΛN sup k∈{0,1,...,Aℓd

∗}

G(k,ℓ,n_x) =0 , (6.6)

wherenx:=|Γx,α,ℓ|,νk,nis the measure on{ζ∈Nn: Pn

i=1ζ(i) =k}obtained by taking the product measure with the same marginals asνρ∗ and then conditioning on the event that the total number

of particles equalsk, and where G(k,ℓ,n):=νk,n

h ℓ−d

∗ n X

i=1

g(ζ(i))−m_αφ(k/m_αℓd_∗) i

. One can prove that forQ–a.a.ωit holds

lim sup N↑∞,ℓ↑∞,L↑∞

Avx∈ΛNI(|nx/ℓ d

∗−mα|> δ) =0 , (6.7)

for each positive constantδ >0. As in[KL][Chapter 5], one has that lim

ℓ↑∞(k,nsup)∈J

νk,n h

n−1 n X

i=1

g(ζ(i))−φ(k/n) i

=0 . (6.8)

where

J :=n(k,n)∈N2 : k

ℓd ∗

6A, n

ℓd ∗

∈(m_α−δ,(m_α+δ)∧1)o. At this point (6.6) follows from (6.7) and (6.8).

A

Zero range process on

Z

d

with random conductances

Recall that the enviroment ω= ω(b) : b ∈E_d is given by a family of i.i.d. random variables parameterized by the setE_d of non–orientied bonds inZd, d >2. We denote by Qthe law of ω, we assume thatQ(ω(b) ∈ [0,c₀]) = 1 and that Q(ω(b) > 0) is supercritical. We fix a function g:N→[0,∞)as in Subsection 2.2. Given a realization ofω, we consider the zero range process

η(t)onZd whose Markov generator N2L acts on local functions as N2Lf(η) =N2X

e∈B X

x∈Zd

g(η(x))ω(x,x+e)€f(ηx,x+e)−f(η)Š, (A.1)

whereB={±e1,±e2, . . . ,±ed},e1, . . . ,ed being the canonical basis ofZd. Given anadmissible ini-tial distribution ¯µN on{0, 1}Zd (i.e. such that the corresponding zero range process is well defined), we denote byP_{ω, ¯µ}N the law of(η_t : t>0). Trivially, the zero range process behaves independently on the different clusters of the conductance field.

IfQ(ω(b)>0) =1, thenQ–a.s. the infinite clusterC(ω)coincides withZd and the hydrodynamic behavior of the zero range process onZd is described by Theorem 2.1. IfQ(ω(b)>0)<1 the bulk

behavior of the zero range process onZd is different due to the presence of finite clusters acting as traps, as we now explain.

First, we observe that the finite clusters cannot be too big. Indeed, as byproduct of Borel–Cantelli Lemma and Theorems (8.18) and (8.21) in[G], there exists a positive constantγ >0 such that for

Q–a.a.ωthe following property (P1) holds:

(P1) for each N >1 and each finite cluster C intersecting the box [−N,N]d_{, the diameter ofC is} bounded byγln(1+N).

Lemma A.1. Suppose thatωhas a unique infinite clusterC(ω)and thatωsatisfies the above property (P1). Let G∈C_c(Rd)andη∈NZd be such thatη(x) =0for all x ∈ C(ω). Call∆_G the support of G and call ∆¯_G the set of points z ∈Rd having distance from ∆_G at most1. Then there exist positive constants N₀(G,γ), C(G,γ) depending only on G and γsuch that the zero range process onZd with initial configuration η satisfies a.s. the following properties: ηt(x) = 0 for all x ∈ C(ω) and, for N>N0(G,γ),

N

−d X x∈Zd

G(x/N)η(x)−N−d X x∈Zd

G(x/N)η_t(x)

6C(G,γ)

ln(1+N) Nd+1

X

x∈Zd_:x/N∈∆¯ G

η(x). (A.2)

We point out that the zero range process is well defined when starting inη, indeed the dynamics reduces to a family of independent zero range processes on the finite clusters, while the infinite clusterC(ω)remains empty.

Proof. The fact thatηt(x) =0 withx 6∈ C(ω)is trivial. Let us prove (A.2). Without loss of generality we can suppose thatG has support in[−1, 1]d _{(the general case is treated similarly). Let us write} C₁N,C₂N, . . . ,C_kN

N for the family of finite clusters intersecting the box[−N,N]

d_{. For each clusterC}N i we fix a point xN_i ∈C_iN. Since by the property (P1) eachC_iN has diameter at most γln(1+N), we have

|G(x/N)−G(x_iN/N)|6C(G)γln(1+N)/N, ∀i: 16i6kN, ∀x∈CiN. The above estimate implies that

N

−d X x∈Zd

G(x/N)ηt(x)−N−d kN X

i=1

G(xN_i /N) X x∈CN

i

ηt(x) 6

C(G)γN−d−1ln(1+N) kN X

i=1 X

x∈CN i

ηt(x).

Using now that the number of particles in each cluster is time–independent and that for N large enoughC_iN ⊂[−2N, 2N]for alli=1, . . . ,k_N, we get the thesis.

As a consequence of Lemma A.1, ifωhas a unique infinite cluster and ifωsatisfies property (P1), then for any admissible initial configurationη0 (i.e. such that the zero range process onZd is well defined when starting inη0) and anyG∈Cc(Rd), it holds

N−d X x∈Zd

G(x/N)η_t(x) =N−d X x∈C(ω)

G(x/N)η_t(x) +N−d X x6∈C(ω)

where|o(1)|6C(G,γ)ln(1+N) Nd+1

P

x∈Zd_:x/N∈∆¯

Gη(x). At this point, denoting by ¯µ

N _{the initial} distribu-tion of the zero range processηtonZd, one can derive the hydrodynamic limit ofηtif the marginals of ¯µN onC(ω)andZd\ C(ω), respectively, are associated to suitable macroscopic profiles. In what follows, we discuss a special case where this last property is satisfied.

We fix a smooth, bounded nonnegative functionρ0 :Rd →[0,∞)and for each N we define ¯µN as the product probability measure onNZd such that for all x∈Zd it holds

¯

µN η(x) =k

=ν_ρ0(x/N)(η(x) =k), (A.4)

whereν_ρis defined as in Subsection 2.2 with the difference that now it is referred to allZd and not only toC(ω).

We callµN the marginal of ¯µN onC(ω): µN is a product probability measure onNC(ω) satisfying (A.4) for all x ∈ C(ω)(note thatµN depends onω). Similarly, we callν_ρ,C(ω)the marginal ofν_ρ

onC(ω). By the discussion at the end of Section 2, if the smooth profileρ0converges sufficiently fast at infinity to a positive constantρ∗, it holds

lim sup N↑∞

N−dH(µN|ν_ρ∗,C(ω))<∞ Q–a.s. (A.5)

Theorem A.2. Suppose that the bounded smooth profileρ0:Rd→[0,∞)satisfies(A.5). Then for all t>0, G∈Cc(Rd)andδ >0, forQ–a.a. ωit holds

lim N↑∞Pω, ¯µ

N

N

−d X x∈Zd

G(x/N)ηt(x)− Z

Rd

G(x)ρ(x,t)d x > δ

=0 (A.6)

where, setting m=Q(0∈ C(ω)),

ρ(x,t) =mρ˜(x,t) + (1−m)ρ0(x) (A.7) andρ˜:Rd×[0,∞)→Ris the unique weak solution of the heat equation

∂tρ˜=∇ ·(D∇φ(ρ˜)) (A.8)

with boundary conditionρ˜0=ρat t=0.

Proof. Since ¯µN is a product measure and the dynamics on different clusters is independent, the process restricted toC(ω)has law P_ω,µN. As discussed at the end of Section 2, to this last process we can apply Theorem 2.1. Since, for Q-a.a. ω the initial distributionsµN are associated to the macroscopic profilemρ0, we conclude that forQ-a.a. ωit holds

lim N↑∞Pω, ¯µ

N

N

−d X x∈C(ω)

G(x/N)ηt(x)−m Z

Rd

G(x)ρ˜(x,t)d x > δ

=0 . (A.9)

Let us now consider the evolution outside the infinite cluster. Let us write N−d X

x6∈C(ω)

G(x/N)η0(x) =N−d X

x∈Zd

G(x/N)η0(x)−N−d X

x∈C(ω)

We know that, whenηis sampled with distribution ¯µN, the addenda in the r.h.s. converge in prob-ability toRG(x)ρ0(x)andm

R

G(x)ρ0(d x), forQ–a.a.ω. As a consequence the l.h.s. converges in probability to(1−m)R G(x)ρ0(x)forQ–a.a.ω. In addition,

sup N>1

nZ ¯

µN(dη)N−d X x:x/N∈∆¯G

η(x)o<∞.

The above observations and Lemma A.1 (cf. (A.3)) imply that lim

N↑∞Pω, ¯µ N

N

−d X x6∈C(ω)

G(x/N)η_t(x)−(1−m) Z

Rd

G(x)ρ0(x)d x > δ

=0 . (A.10)

The thesis then follows from (A.9) and (A.10).

Acknowledgements. The author kindly acknowledges the Department of Mathematics of the Uni-versity of L’Aquila for the kind hospitality while part of this work was being done. She also thanks the anonymous referee for suggesting the problem discussed in Appendix A.

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