Diffusive Long–time Behavior of Kawasaki Dynamics

If P t is the semigroup associated with the Kawasaki dynamics on Z d and f is a local function on the configuration space, then the variance with respect to the invariant measure µ of P t f goes to zero as t → ∞ faster than t −d/2+ε , with ε arbitrarily small. The fundamental assumption is a mixing condition on the interaction of Dobrushin and Schlosman type. 1 With the support of Marie-Curie fellowship of the EC program " Human Potential " , contract number HPMF-CT-2001-01177.


Introduction
Consider a Markov process (η t ) t≥0 taking values in an infinite product space Ω := S Z d , so that η t = (η t (x)) x∈Z d .S could be, for instance, a subset of N, and the variable η t (x) is sometimes thought of as the number of particles at x at time t.Assume then that this process is reversible with respect to a probability measure µ on Ω which is a Gibbs measure for some interaction J.A very general problem for this class of processes is the study of the relationships between the properties of the interaction J and the behavior of the process.Among the infinite possible ways of constructing Markov processes of this type we single out two special categories: (1) spin-flip processes2 also called Glauber dynamics, and (2) (nearest neighbor) particle-exchange processes or Kawasaki dynamics.In the first case we have that the coordinates of η t can change only one at a time.More precisely, if L G is the generator of the process, then and c(η, η ) = 0 unless η and η differ in exactly one coordinate.Similarly, in a Kawasaki dynamics the transition rate c(η, η ) is zero unless η can be obtained from η by transferring one particle from x to y where x, y are nearest neighbors in Z d .
One of the most important result concerning Glauber dynamics is a theorem asserting the equivalence between a mixing condition of Dobrushin and Shlosman type [DS87] on the interaction J and the fact that the distribution of η t converges exponentially fast to the invariant measure µ in a rather strong sense.For a comprehensive account on this subject we refer the reader to the beautiful review paper [Mar99].For Kawasaki dynamics, which we study in this paper, the situation is more complicated.In fact, even if the interaction J is zero and consequently µ is a product measure, the process is nevertheless an "interacting" (i.e.non-product) process.These type of processes are also called "conservative dynamics" because if we run them in a finite volume Λ ⊂ Z d then the function t → N Λ (η t ) := x∈Λ η t (x) is constant.The specific problem we want to address is: let (P t ) t≥0 be the semigroup associated with the process, and let f be a real function on Ω.How fast does the quantity P t f converges to the expectation µ(f ) := Ω f dµ?Of course there are several ways of interpreting this convergence, however, since µ is supposed to be a reversible measure, one of the most natural quantities to study is the L 2 (µ) distance.Hence we are looking at the long-time behavior of the quantity where Var µ stands for the variance w.r.t.µ.One of the first things to realize is that the constraint imposed by the conservation law prevents this convergence from being exponentially fast even when J = 0.It is fairly easy to show (see [Spo91], pag.175-6) that the quantity (1.1) (with, say f (η) := η(0)) cannot be smaller than C t −d/2 .Actually t −d/2 is conjectured to be the correct long-time asymptotics, when (J is such that) µ is somehow close to a product measure, this conjecture being hatched from the idea that these processes are discretized versions of diffusions in R d .For elliptic diffusions, a standard way of proving the t −d/2 decay is [Dav89, Sect 2.4] by means of the so called Nash inequalities stating where E is the Dirichlet form given by Unfortunately, in the (morally) infinite dimensional framework of Kawasaki dynamics, this approach has been successful only for a special model called symmetric simple exclusion process [BZ99a,BZ99b] where S = {0, 1} and the invariant measure µ is Bernoulli.
It is clear, on the other side, that a piece of information that should be relevant for this problem is the fact that, while the generator of the process has no spectral gap in the infinite volume, if we denote with L the generator in Z d ∩ [− , ] d then we have for large [LY93,CM00b] gap(L ) ∼ C −2 . (1.2) A new approach was then developed in [JLQY99] where thanks to a combination of (1.2) with techniques imported from the hydrodynamic limit theory it was proved that, for the symmetric zero-range process one has where C(f ) is an explicit quantity.More recently [LY03] the same result (apart from logarithmic corrections) has been extended to the Ginzburg-Landau process with a potential which is a bounded perturbation of a Gaussian potential.Both the zero-range and the Ginzburg-Landau process have an invariant measure which is a product measure.The first results which apply to a process with a non-product invariant measure µ were obtained in [CM00b].Their main assumption is a mixing condition on µ.In that paper it has been shown in a very simple way that (1.2) supplemented with a soft spectral theoretic argument implies an almost optimal upper bound when d = 1, 2. More precisely for all ε > 0 and for all local function f on Ω, there exists C ε,f > 0 such that (1.4) Their strategy, appealing for its simplicity, seems however unable to yield the correct asymptotics in more than two dimensions.
In the present paper we extend inequality (1.4) to arbitrary values of d, following the original approach of [JLQY99] which the authors predicted would be powerful enough to treat processes with non-trivial invariant measures.We stress, however, that we are unable to prove the sharper equality (1.3).The main reason is that we have been incapable of extending the very precise "hydrodynamical" estimates of [JLQY99, Section 5] to the type of processes we consider in this paper, in which the invariant measure is only assumed to satisfy a certain mixing condition.
Acknowledgements.We warmly thank Fabio Martinelli for many helpful suggestions and continuous support, and Lorenzo Bertini and Claudio Landim for many interesting conversations.

Lattice and configuration space
The lattice.We consider the d dimensional lattice Z d whose elements are called sites x = (x 1 , . . ., x d ) and where we define the norms We denote by E Λ the set of all edges with both endpoints are in Λ and by E Λ the set of all edges with at least one endpoint in Λ.Given Λ ⊂ Z d we define its interior and exterior n-boundaries as respectively, A is a partition of Λ, i.e.Λ is the disjoint union of the elements of A.
The configuration space.Our configuration space is Ω = S Z d , where S = {0, 1}, or Ω V = S V for some V ⊂ Z d .The single spin space S is endowed with the discrete topology and Ω with the corresponding product topology.Given η ∈ Ω and Λ ⊂ Z d we denote by η Λ the restriction of η to Λ.If U , V are disjoint subsets of Z d , σ U τ V is the configuration on U ∪ V which is equal to σ on U and τ on V .We denote by π x the standard projection from Ω onto S, i.e. the map η → η(x).
If Λ ∈ F, N Λ stands for the number of particles in Λ, i.e.N Λ = x∈Λ η(x).If A is a collection of finite subsets of Z d , we define N A as We also define the σ-algebras When Λ = Z d we set F = F Z d and F coincides with the Borel σ−algebra on Ω with respect to the topology introduced above.
If f is a function on Ω, S f denotes the smallest subset of Z d such that f (η) depends only on η S f .f is called local if S f is finite.We introduce 3 operators (1) the translations: (2) the spin-flip: The capitalized versions of s x , t xy act on functions in the obvious way The Glauber and Kawasaki "gradients" are then respectively defined as We denote with f u the supremum norm of f , i.e. f u := sup η∈Ω |f (η)| and with osc(f ) the oscillation of f , i.e. osc(f ) := sup f − inf f .

The interaction and the Gibbs measures
In the following we consider a translation invariant, summable interaction J, of finite range r, i.e. a collection of functions J = (J A ) A∈F , such that J A : Ω → R is measurable w.r.t.F A , and Conditions (H1), (H2), (H3) will always be assumed without explicit mention.The Hamiltonian (H Λ ) Λ∈F associated with J is defined as ) and τ is called the boundary condition.For each Λ ∈ F, τ ∈ Ω the (finite volume) Gibbs measure on (Ω, F), are given by where Z τ Λ is the proper normalization factor called partition function, and 1I is the indicator function.In the future we are going to consider an interaction J with an explicit additional chemical potential λ.In particular we will consider a chemical potential on a polycube (Λ, , A) such that λ is constant in each cube x + Q ∈ A. For this reason, given such a polycube, and given λ ∈ R A we define (2.4) The associated finite volume Gibbs measures are denoted by µ τ A,λ .Given a bounded measurable function f on Ω, µ τ Λ f denotes expectation of f w.r.t.µ τ Λ , while, when the superscript is omitted, µ Λ f stands for the function σ → µ σ Λ (f ) which is measurable w.r.t.F Λ c .Analogously, if X ∈ F, µ V (X) := µ V (1I X ).µ(f, g) stands for the covariance (with respect to µ) of f and g.The variance of f is (accordingly) denoted by µ(f, f ) or, alternatively, by Var µ (f ).The set of measures (2.3) satisfies the DLR compatibility conditions Our main assumption on the interaction J is an exponential mixing property for the finite volume Gibbs measures µ τ A,λ , uniform in the chemical potential λ.More precisely we assume that: (USM) There exist Γ 0 , m, 0 ∈ (0, ∞), and for every local function f on Ω there is A f > 0 which depends only on |S f | and f u , such that for all polycubes (Λ, , A) with ≥ 0 for all pairs of local functions f, g we have Condition (USM) is easily implied, except for the uniformity in λ, by the Dobrushin and Shlosman's complete analyticity condition (IIIc) in [DS87].As for the necessity of assuming this uniformity in λ we refer the reader to the remark after the "Definition of property (USMT)" in [CM00b].
By standard arguments it is not hard to check that (USM) implies that there exists This inequality becomes effective when S f and S g are "far apart" enough, in which case it can be written in a simpler form.More precisely there exists Γ 1 = Γ 1 (d, r, J , Γ 0 , m) such that if (2.10) From (2.6) the following well known fact easily follows Proposition 2.1.Under hypothesis (USM) there is exactly one Gibbs meassure for J which we denote with µ.
We introduce the (multi-)canonical Gibbs measures on (Ω, F): let (Λ, , A) be a polycube 3 and let M = (M V ) V ∈A be a possible choice for the number of particles in each cube V ∈ A, i.e.
Then we define (remember (2.1)) (2.12) We have, for f ∈ L 1 (µ) in this way we can write the "multicanonical DLR equations" as In the special case where A = {Λ} consists of a single element we (slightly) improperly write 3 multi-canonical measures can obviously be defined on an arbitrary partition of Λ

The dynamics
We consider the so-called Kawasaki dynamics, a Markov process with generator L V , where V Z d and We denote by L τ V,N the operator L V acting on L 2 (Ω, ν τ V,N ).Assumptions (1), ( 2) and (3) guarantee that there exists a unique Markov process whose generator is L τ V,N , and whose semigroup we denote by (P V,N,τ t ).With (P t ) t≥0 we denote the infinite volume semigroup which is reversible w.r.t.µ, while L stands for the generator of (P t ) t≥0 .
A fundamental quantity associated with the dynamics of a reversible system is the spectral gap of the generator, i.e.
When Q equals the unique infinite volume Gibbs measure µ, we (may) omit it as a subscript.Analogously we omit the subscript V when V = Z d , so we let for simplicity The Dirichlet form associated with the generator L τ V,N is then given by E ν τ V,N ,V (f ).The gap can also be characterized as (2.16)

Main result
Constants.Throughout this paper we tacitly assume to have chosen once and for all a value of the dimension d of the lattice Z d , an interaction J of finite range r satisfying (H1), (H2), (H3), a set of transition rates (c e ) e∈E Z d satisfying (K1), (K2), (K3).Our main result and most of the results contained in this work hold when the interaction J is such that the mixing hypothesis (USM) is also satisfied.With the word "constant" we denote any quantity which depends solely on the paramenters which have been fixed by means of these hypotheses, namely d, r, J , c m , c M , Γ 0 , m, 0 .Analogously "for x large enough" means for x larger than some constant.For simplicity we write things like "Assume (USM).Then for all ε > 0 there is C > 0 such that . . ." without reiterating that C depends not only on ε, but in principle, on all the paramenters mentioned above.
Theorem 2.2.Assume (USM).Then for all ε > 0 and for all local functions f on Ω there is (a) This result has been proved for d = 1, 2 in [CM00b], so we are going to consider only the case d ≥ 3.
(b) One might want to be more ambitious and study, instead of the quadratic fluctuations of P t f , the convergence to the invariant measure in some stronger sense, say L ∞ (µ).We refer the reader to the introduction of [JLQY99] where it is explained how, for these kind of models, the long time behavior of the quantity |P t f (η) − µ(f )| has a nontrivial dependence on the starting point η, which makes pointwise estimates a much harder problem.
3 Outline of the proof of Theorem 2.2 Let d ≥ 3, let, as usual, µ be the unique infinite volume Gibbs measure for the interaction J, and define f, g := µ(f g), f := µ(f 2 ) 1/2 .Let f be a local function such that µf = 0, let f t := P t f and let K t := √ t .In the following it will be convenient to average over spatial translations, hence we define Then we write The second term in (3.1) is by far the easier.In fact, since P s is a contraction in L 2 (µ), we have, for all s, t > 0 so, using our mixing assumption (2.10), we obtain that there is 3) is implied, by iteration, by the following statement In order to prove (3.4) we write, using (3.2) We claim that in order to prove (3.4) it is sufficient to show that for some and, using (3.5), we find Hence the theorem follows from (3.6).

Proof of statement (3.6)
Let K ≤ K t , let (B L 2 , , A) be a polycube, and choose two more integers L, L 1 such that ≤ L < L 1 < L 2 .We anticipate5 that we are going to choose and define Thanks to translation invariance, we have, since µ g x,t = µf = 0, For simplicity we define the following orthogonal projections in L 2 (µ) Then, since (3.9) On the other side, since (3.10) From (3.9), (3.10) we get where Var(f | •) stands for the conditional variance of f (w.r.t.µ).We now proceed to estimate each of the four terms in (3.11) and we are going to prove that (3.6) holds.
First term in (3.11).In Section 4 we generalize the so-called "cutoff estimate" (Proposition 3.1 of [JLQY99]) to the case where the measure µ is no longer a product measure, but it satisfies our mixing condition (USM).The result is (more or less) the same as in [JLQY99].
Proposition 3.1.Assume (USM).Then there exists C > 0 such that, for all local functions g on Ω, for all t ≥ 1 such that S g ⊂ B 3 √ t , and for all L ∈ Z + , we have (3.12) If we apply this result to the first term of (3.11) we find, for all x ∈ B L , Second term in (3.11).This term keeps track of the fluctuation of the number of particles in the various blocks which make up the polycube (Λ 2 , , A).We use the following result whose proof appears in section 7. The integral of the second term in (3.11) can be estimated as follows: Proposition 3.2.Assume (USM).Then, for all ε > 0, for all local function f on Ω, there exists A = A(f, ε) such that: for all polycubes (Λ, , A) for all positive integers K, L, taking into account definitions (3.7), and for all t > 0 we have Third term in (3.11).Since Var(f ) ≤ osc(f ) 2 /2, we get In order to estimate the difference appearing in the RHS of (3.15) we use the following result which will be proved in Section 5 (see Corollary 5.7): Proposition 3.3.Assume (USM).Then there exists C > 0 such that for all polycubes (B L , , A), for all functions f on Ω such that S f ⊂ B L , and for all M ∈ M A , we have From (3.15) and (3.16) (applied to the polycube (Λ 2 , , A)) and thanks to the fact that Fourth term in (3.11).In Section 6 we prove a Poincaré inequality for the multicanonical measure, more precisely Proposition 3.4.Assume (USM).Then for all γ ∈ (0, (d − 1) −1 ) there exists C γ such that for all polycubes (B L , , A) with L ≤ 1+γ we have, for all local functions If L 2 ≤ 1+γ 0 we can apply Proposition 3.4 to our polycube (Λ 2 , , A).In this way we can estimate the fourth term in (3.11) as In order to find a suitable upper bound to the Dirichlet form appearing in the RHS of (3.19) we proceed as follows: given an edge e of Z d we have, for all f ∈ L 2 (µ) Thanks to Proposition 3.1 (applied to the sigma-algebra F B L 1 ), and using the fact that S gx ⊂ B L (S g ), we obtain that for some constant C 1 From (2.14), (3.19), (3.20) we get that there is (3.21)For any zero mean function f in the domain of E we have End of proof of Theorem 2.2.To conclude the proof we choose appropriate values for , L, L 1 , L 2 and collect the various pieces together.Choose then a real number α such that 5 C 3 α 2 < 4 −d/2 , and let6  4 Cutoff estimate and proof of Proposition 3.1 In this section we prove Proposition 3.1.We observe that the factor 3 appearing in the assumption S g ⊂ B 3 √ t is completely arbitrary.By redefining the constant C one can replace this 3 with any number.We follow the strategy of Proposition 3.1 in [JLQY99], with suitable modifications required by the fact that, in our case, µ is not a product measure.
Lemma 4.1.Assume (USM) and let, for j ∈ N, Then, there exists a constant C > 0 such that for all ϑ > 0, and for all local functions g on Ω, we have (remember (2.15)) Proof.We let f, g := µ(f g), f := µ(f 2 ) 1/2 , and we define, for A straightforward computation shows that if e = {x, y} ⊂ B j then we have (remember (2.2)) In the special case in which e ⊂ B j−r we have h j e = 1, thus, (4.1) reduces to If instead e = {x, y} ∈ δB j then the formula is slightly more complicated.Assume x ∈ B j , y ∈ B c j , and let q e (σ) = 1I {σ(x) =σ(y)} .Then where

It is easy to verify that
A j V j xy = A j W j xy = 0 .

By definition we have
µ c e ∇ e (A j g)∇ e g , so, letting, Y e := µ c e ∇ e (A j g)∇ e g , we can write where For what concerns those edges e ⊂ B j−r which contribute to X 1 we observe that c e A j (f ) = A j (c e f ).From this equality, from the fact that A j is an orthogonal projection in L 2 (µ), and from (4.2), it follows that, for each e ⊂ B j−r In order to estimate X 2 we use (4.1) and we get (c m , c M are the minimum and maximum transition rates) Using xy ≤ (ϑx 2 + ϑ −1 y 2 )/2 in the second term, we get Since J < ∞, there exists C 0 > 0 such that for all edged e and all sites x In this way we obtain The term X 3 can be estimated using (4.3) and (4.5) as where C 1 is some positive constant.Collecting the terms in (4.4), (4.6), (4.7), we find that there exists a constant C 2 > 0 such that In order to estimate the three sums which appear in (4.8) we use the following elementary Hilbert space inequality.
Proposition 4.2.Let (V, •, • ) be a Hilbert space, and let (u i ) n i=1 be a finite sequence of elements of V .Define Then, for all v ∈ V , we have and the result follows.
Consider now the first sum in the RHS of (4.8) The idea is to use Proposition 4.2 with f, g replaced by A j (f g).Thanks to the hypothesis (USM) on the interaction J, there exists a constant C 3 > 0 such that max e ∈D j e∈D j |A j (U j e U j e )| ≤ C 3 ∀j ∈ Z + .(4.10)By consequence, using (4.9), (4.10), the fact that U j e is measurable w.r.t.F B j+r and that A j (U j e ) = 0, we get (4.11) From (4.8), (4.11) and the analogous inequalities for the terms Lemma 4.1 follows.

End of proof of Proposition 3.1
Once we have estabilished Lemma 4.1, Proposition 3.1 follows more or less in the same way as in [JLQY99].We include the argument for completeness.
Let , L be two positive integers, let ϑ > 0, and let α i := e i/(ϑC) for i ∈ N, where C is the constant which appears in Lemma 4.1.We assume C ≥ 1 otherwise we redefine C as C ∨ 1.Given g ∈ L 2 (µ) we also let g t := P t g.Define then the function and notice that it can be also written as Differentiating and using to Lemma 4.1 we obtain Using the summation by parts formula we can rewrite F (t) as With our choice for α i , we have that ϑC( By consequence F (t) ≤ F (0)e 2t/ϑ 2 , so, if S g ⊂ B 2 r we have Choosing now = 3( √ t/r +1) and ϑ = √ t/(2rC), we obtain Proposition 3.1.

Influence of the boundary condition on multicanonical expectations
In this section we study how the multicanonical expectation ν τ A,M (f ) of a function f on Ω is affected by a variation of the boundary condition τ .More precisely we want to find an upper bound to the quantity This problem has been studied, in a particular geometrical setting, in [CM00a].Following a similar approach we are going to show how to deal with a more general geometry.
Let then (Λ, , A) be a polycube and let M ∈ M A a possible choice for the number of particles in each element of A. In order to study how the quantity ν τ A,M (f ) depends on τ , we first approximate this multicanonical expectation with a grandcanonical expectation µ τ A,λ (f ) in which λ is a suitable chemical potential (remember (2.4)) which we assume constant in each cube of A. The value of λ is determined by the requirement that the expectation of the number of particles in each cube is equal to the number of particles fixed by the multicanonical measure.In other words we want µ τ A,λ (N V ) = M V for all V ∈ A. The existence of this tilting field λ is proved in the appendix of [CM00a].Thus there is a map λ : (A, M, τ ) → λ such that For brevity we define μτ A,M := µ τ A, λ(A,M,τ ) .

The basic estimate
The idea for estimating (5.1) is to write (5. 3) The first and third term can be estimated using Proposition 5.1 below, a result concerning the "equivalence of the ensembles", while the second term will be taken care of in Proposition 5.2.
Proposition 5.1.Assume (USM).There exists C > 0 such that for all polycubes (Λ, , A), for all M ∈ M A , for all functions f on Ω such that |S f | ≤ d/2 , we have (5.4) where Proof.It is a straightforward consequence of Theorem 5.1 in [CM00a] (see aso Theorem 4.4 in [BCO99]).We just observe that the "bad block" estimate in that theorem is good enough for our purposes.
Proposition 5.2.Assume (USM).There exist ζ, C > 0 such that for all polycubes (Λ, , A), for all M ∈ M A , for all functions f on Ω such that S f ⊂ Λ we have where Proof.The proof of this statement requires some modifications of the proof of Proposition 7.1 in [CM00a], where a different geometry is considered.We first observe that we can assume otherwise there is nothing to prove.For simplicity we enumerate (in an arbitrary way) the set (5.7) and we let M i := M Λ i .Let λ 0 , λ 1 ∈ R n be defined by (5.8) If we denote by h the Radon-Nikodym density of µ σ Λ,λ 1 with respect to µ τ Λ,λ 1 , i.e.
(5.9) we can write (5.10) The covariance term in the RHS of (5.10) can be bounded using (2.10).In fact we have and, using inequalities (5.6) it is easy to show that if the constant ζ is chosen small enough then condition (2.9) is satisfied.Hence thanks to (2.10) and the fact that µ τ A,λ 1 (h) = 1, we find (5.11) We are now going to show that which, together with (5.11), proves the Proposition.We start by introducing a chemical potential λ s , s ∈ (0, 1), which interpolates between λ 0 and λ 1 Let then, for any local function g, and for i, j = 1, . . ., n, ϕ i (g) := 1 0 µ τ A,λs (N Λ i , g) ds (5.13) (5.15) Then we have, letting and, analogously, (5.17) On the other side we have, by (5.8) and (5.9) by consequence, assuming that B is invertible (we prove it later) we obtain For any n × n matrix A, let A be the norm of A when A is interpreted as an operator acting on (R n , | • | ∞ ), i.e.
We turn then our attention to the two factors in the RHS of (5.24).In order to obtain proper bounds on them we finally choose the set I and complete our definition of the matrix G in (5.28).We define (5.30) At this point we would like to say that d(Λ i , W 0 ) is roughly equal to κ(i, I) .More precisely we have Lemma 5.4.For all i = 1, . . ., n we have Proof of the Lemma.Let's start with the lower bound on d(Λ i , W 0 ).We have which, when i ∈ I can be improved as which is trivially true also when i ∈ I.For the upper bound we observe that, if x ∈ Λ j for some j ∈ I, then Thus, if we let Λ I := ∪ j∈I Λ j , we have that Λ I ⊂ B /3+ −1 (W 0 ), so ). Thanks to Lemma 5.3 we can write (5.33) (5.34) If instead i ∈ I (and is large enough) we can use (2.10) and we get which, together with (5.31) yields, for large enough, (5.35) From (5.34), (5.35), we finally get (5.36) ) and end of the proof.In order to prove (5.12) we have to bound the last factor in (5.24), namely |G −1 ϕ(f )| 1 .We have Observe first that using Lemma 5.4, If i is such that d(Λ i , S f ) ≤ /3 we use inequality (4) of Lemma 5.3 and we find where we have set where A 2 := A 1 (8/3) d .In order to estimate the contribution of those terms with d(Λ i , S f ) > /3 we use (2.10) and, again, (5.38) and we obtain Furthermore it is easy to see that, if is large enough, then (5.41) From (5.40), (5.41) it follows that which, together (5.39) implies with A 3 := A 1 + A 2 .Finally from (5.24), (5.36) and (5.43), inequality (5.12) follows, and the proof of Proposition 5.2 is completed.

Improving the basic estimate
Inequality (5.44) can be iterated and consequently improved.What follows is a generalization of the strategy adopted in [CM00a].
Proposition 5.6.Assume (USM).There exists C > 0 such that the following holds: let (Λ, , A) be a polycube, and let f be a function on Ω such that S f ⊂ Λ.
Given a pair of boundary conditions τ, σ ∈ Ω, let W := {x ∈ ∂ + r Λ : τ (x) = σ(x)}.Assume that there exists an increasing sequence of polycubes (T k , , A k ), k = 0, . . ., n, such that Then, for all M ∈ M A , (5.45) We denote with M k the restriction of M to A k .The function f k is measurable w.r.t.F ∂ + r T k ∩Λ .Denoting with Ω the set of all η ∈ Ω such that η Λ c = τ Λ c , we can write Since, by hypothesis, d(T k , W ) ≥ d(T n , W ) > r, we have By consequence (remember that osc(f ) := sup f − inf f ) (5.47) On the other side, if we let (5.48) Define now a set Tk−1 slightly larger than T k−1 , such that f k−1 is measurable w.r.t.F Tk−1 .We let Tk−1 := B (T k−1 ) ∩ Λ.The reason for taking the -boundary of T k−1 instead of the r-boundary, which would be enough for the measurability requirement, is that, in this way there exists Âk−1 ⊂ A k , such that ( Tk−1 , , Âk−1 ) is a polycube.The RHS of (5.48) can be estimated as (5.49) The idea, at this point, is to bound the last factor in the RHS of (5.49) using inequality (5.44).Thanks to hypothesis (iii) the distance between S h x k and Tk−1 can be bounded from below as ) .So we can apply Corollary 5.5, and, since the uniform norm of h x k is at most exp(4 J ), we obtain, with a suitable redefinition of the constant C, (5.50) From (5.46), (5.47), (5.48), (5.49), (5.50), it follows that On the other side, since by hypothesis S f ⊂ T 0 , we have osc(f 0 ) ≤ osc(f ) ≤ 2 f u , and the Proposition is proved.
In the following Corollary we consider a particular situation where previous result can be applied and we write down a more explicit expression for the RHS of (5.45).
Proof.Let j be the smallest integer such that S f ⊂ (B j + y) ∩ Λ for some y ∈ Z d .Inequality (5.51) follows (after a redefinition of C) from Proposition 5.6, if one takes where n = d(S f , W )/[(3d + 4) ] − 2.

Poincaré inequality
In this section we prove Proposition 3.4.Our goal is to obtain a Poincaré-type inequality for the multicanonical measures ν τ A,M on the polycube (B L , , A) when L ≤ 1+γ with γ < (d − 1) −1 .This restriction on γ is really fictitious and springs from the fact that the quantity appearing in brackets in the RHS of (5.51) must be much smaller than one.In order to overcome this difficulty one could iterate inequality (5.51) again and obtain a result suitable for larger values of L.
We also observe that (3.18) is weaker than the standard Poincarè inequality associated with the measure ν τ A,M , for two reasons: first of all the inequality (3.18) is averaged with respect to the infinite volume measure µ, and, moreover the Dirichlet form in the RHS of (3.18) contains also those terms (∇ xy f ) 2 in which x and y belong to different cubes of A. This weaker inequality is anyway sufficient for our purposes.
Before starting with our proof, we want to remark that an inequality somehow close to the one we are trying to demonstrate requires basically no effort7 .Let in fact A 0 , A 1 be two partitions of On the other side the canonical measure satisfies (see [LY93] and [CM00b]) a Poincaré inequality which says As an aside, we observe that by taking the expectation of (6.2) we get Unfortunately this inequality is not sufficient for our purposes, and the rest of this section is devoted to eliminating the factor γ from the RHS of (6.3).
We use the iterative approach which was introduced in [Mar99].We let δ = 3(3d + 4) and, following [BCC02], we define a sequence of exponentially increasing length scales w k := 4δ (3/2) k/d k = 0, 1, 2, . . .(6.4) Our choice of δ represents the minimum distance which yields an exponent equal to 1 in the RHS of (5.51).Then we define R k as the set of all Λ in Z d such that By consequence we can iterate (6.13) up to k = k 1 and obtain an upper bound for the Poincaré constant of the polycube (B L , , A) as Since γ < (d − 1) −1 from the definition of α if follows that there exists 0 (γ) > 0 such that if ≥ 0 (γ) then α is bounded by a negative power of , hence k 1 α 1 ≤ 1.On the other side there exists K(d) such that On the other side if < 0 (γ) we can simply use (6.2) and obtain hence (3.18) holds if we redefine C γ suitably.

Fluctuations of the number of particles
We prove here Proposition 3.2.Consider a polycube (Λ, , A) and fix ε > 0. For all M ∈ M A , all x ∈ B L and all t ≥ 0, let Then, by reversibility and translation invariance, if s ≥ 0 Thus, using the Cauchy-Schwarz inequality and the invariance of B L under the mapping x → −x, we can write In the next two Lemmas we deals with µ(d e f h M x,s ) 2 .In the proof it will be clear why, at the very begining, we have subtracted R K f .This leads to having d e f instead of f in (7.2).
Lemma 7.1.Assume (USM).For any α > 0, there is C α > 0 such that for all local functions f on Ω with S f 0, for all u ∈ Z d with |u| 1 = 1, and for all positive integers L, we have one can use Lemma 10.1 of [VY97] where estimate (7.3) is proved when f is the particular function σ(0) and the result follows.
Back to the inequality (7.1).Using Lemma 7.2 together with (7.2) and the fact that |γ y | = |y| 1 ≤ dK for any y ∈ B K , we get that for any M ∈ M A , any s ≥ 0, In the last inequality we used the fact that E We (ϑ x H) = E −x+We (H) for any x and any H, due to the translation invariance property.Then, the bound |γ y | ≤ dK and an explicit computation gives  Proposition 3.2 then follows, after a redefinition of ε, from (7.1), (7.8), (7.9), (7.10), and from the fact that |W e | ≤ (2L + 1) dε .
Remark 7.3.Let us briefly explain the difference with the product case.In that case, the first term in (7.7) is null.By consequence, on can choose the boxes |W e | independent of L and so the logarithmic Sobolev constant used in the Herbst argument is also constant.
i∈{1,...,d} |x i | .The associated distance functions are denoted by d p (•, •) and d(•, •).We define B L as the ball in Z d centered at the origin with radius L with respect to the norm | • |, i.e.B L := {x ∈ Z d : |x| ≤ L}.Let also, for y ∈ Z d , B L (y) := B L + y, and, more generally, for A ⊂ Z d , B L (A) := B L + A = {x ∈ Z d : d(x, A) ≤ L}.If Λ is a finite subset of Z d we write Λ Z d .The cardinality of Λ is denoted by |Λ|.F is the set of all nonempty finite subsets of Z d .Two sites x, y are said to be nearest neighbors if |x − y| 1 = 1.An edge of Z d is a (unordered) pair of nearest neighbors.
.13) The nonnegative real quantities c e (σ) are the transition rates for the process.The general assumptions on the transition rates are (K1) Finite range.c e is measurable w.r.t.F Br(e) .(K2) Detailed balance.For all e ∈ E Z d we have ∇ e c e e −He = 0. (K3) Positivity and boundedness.There exist positive real numbers c m , c M such that c m ≤ c e (σ) ≤ c M for all e ∈ E Z d and σ ∈ Ω.
7.8)for some other constant A .It is well known (see [ABC + 00, Chapter 2] for instance) that for any f , ∂ s Ent(P s f ) ≤ −4E( √ P s f ).Thus, as Ent(P s f ) is non increasing, we have last equality, we have used the definition of entropy.By consequence we have