Logarithmic Sobolev Inequality for Zero-Range Dynamics: independence of the number of particles

We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter L may depend on L but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that this logarithmic-Sobolev constant grows as L^2, that will be presented in a forthcoming paper.


Introduction
The zero-range process is a system of interacting particles moving in a discrete lattice Λ, that here we will assume to be a subset of Z d . The interaction is "zero range", i.e. the motion of a particle may be only affected by particles located at the same lattice site. Let c : N → [0, +∞) be a function such that c(0) = 0 and c(n) > 0 for every n > 0. In the zero-range process associated to c(·) particles evolve according to the following rule: for each site x ∈ Λ, containing η x particles, with probability rate c(η x ) one particle jumps from x to one of its nearest neighbors at random. Waiting jump times of different sites are independent. If c(n) = λn then particles are independent, and evolve as simple random walks; nonlinearity of c(·) is responsible for the interaction. Note that particles are neither created nor destroyed. When Λ is a finite lattice, for each N ≥ 1, the zero-range process in Λ restricted to configurations with N particles is a finite irreducible Markov chain, whose unique invariant measure ν N Λ is proportional to where c(n)! = 1 for n = 0 c(n)c(n − 1) · · · c(1) otherwise.
Moreover the process is reversible with respect to ν N Λ . If the function n → c(n) does not grow too fast in n, then the zero range process can be defined in the whole lattice Z d . In this case the extremal invariant measures form a one parameter family of uniform product measures, with marginals where ρ ≥ 0, Z(α(ρ)) is the normalization, and α(ρ) is uniquely determined by the condition µ ρ [η x ] = ρ (we use here the notation µ[f ] for f dµ).
In this paper we are interested in the rate of relaxation to equilibrium of zero-range processes. In general, for conservative systems of symmetric interacting random walks in a spatial region Λ, for which the interaction between particles is not too strong, the relaxation time to equilibrium is expected to be of the order of the square of the diameter of Λ, as it is the case for independent random walks. On a rigorous basis, however, this result has been proved in rather few cases. For dynamics with exclusion rule and finite-range interaction (Kawasaki dynamics) relaxation to equilibrium with rate diam(Λ) 2 has been proved (see [7,2]) in the high temperature regime. For models without the exclusion rule, i.e. with a possibly unbounded particle density, available results are even weaker (see [5] and [9]). Zerorange processes are special models without the exclusion rule. In some respect zero-range processes may appear simpler than Kawasaki dynamics: the interaction is zero-range rather than finite-range and, as a consequence, invariant measures have a simpler form. However, they exhibit various fundamental difficulties, including: • due to unboundedness of the density of particles various arguments used for exclusion processes fail; in principle the rate of relaxation to equilibrium may depend on the number of particles, as it actually does in some cases; • there is no small parameter in the model; one would not like to restrict to "small" perturbations of a system of independent particles.
In [5] a first result for zero-range processes has been obtained. Let E ν N Λ be the Dirichlet form associated to the zero-range process in Λ = [0, L] d ∩ Z d with N particles. Then, under suitable growth conditions on c(·) (see Section 2), the following Poincaré inequality holds where C may depend on the dimension d but not on N or L. Moreover, by suitable test functions, the L−dependence in (1.3) cannot be improved, i.e. one can find a positive constant c > 0 and functions f = f N,L so that ν N Λ [f, f ] ≥ cL 2 E ν N Λ (f, f ) for all L, N . In other terms, (1.3) says that the spectral gap of E ν N Λ shrinks proportionally to 1 L 2 , independently of the number of particles N . It is well known that Poincaré inequality controls convergence to equilibrium in the L 2 (ν N Λ )-sense: if (S Λ t ) t≥0 is the Markov semigroup corresponding to the process, then for every function f Poincaré inequality is however not sufficient to control convergence in stronger norms, e.g. in total variation, that would follow from the logarithmic-Sobolev inequality The constant s(L, N ) in (1.5) is intended to be the smallest possible, and, in principle, may depend on both L and N .
Our main aim is to prove that s(L, N ) ≤ CL 2 for some C > 0, i.e. the logarithmic-Sobolev constant scales as the inverse of the spectral gap. This is the first conservative system with unbounded particle density for which this scaling is established (see comments on page 423 of [4]). It turns out that the proof of this result is very long and technical, and it roughly consists in two parts. In the first part one needs to show that s(L) := sup N ≥1 s(L, N ) < +∞, (1.6) i.e. that s(L, N ) has an upper bound independent of N , while in the second part a sharp induction in L is set up to prove the actual L 2 dependence. For this induction to work one has to choose L sufficiently large as a starting point, and for this L an upper bound for s(L, N ) independent of N has to be known in advance. Note that for models with bounded particle density inequality (1.6) is trivial. This paper is devoted to the proof of (1.6), while the induction leading to the L 2 growth is included in [3]. The proof of (1.6) is indeed very long, and relies on quite sharp estimate on the measure ν N Λ . After introducing the model and stating the main result in Section 2, we devote Section 3 to the presentation of the essential steps of the proof, leaving the (many) technical details for the remaining sections.

Notations and Main result
Throughout this paper, for a given probability space (Ω, F, µ) and f : Ω → R measurable, we use the following notations for mean value and covariance: and, for f ≥ 0, where, by convention, 0 log 0 = 0. Similarly, for G a sub-σ-field of F, we let µ[f |G] to denote the conditional expectation, the conditional covariance, and the conditional entropy.
If A ⊂ Ω, we denote by 1(x ∈ A) the indicator function of A. If B ⊂ A is finite we will write B ⊂⊂ A. For any x ∈ R we will write x := sup{n ∈ Z : n ≤ x} and x := inf{n ∈ Z : n ≥ x}. Let Λ be a possibly infinite subset of Z d , and Ω Λ = N Λ be the corresponding configuration space, where N = {0, 1, 2, . . .} is the set of natural numbers. Given a configuration η ∈ Ω Λ and x ∈ Λ, the natural number η x will be referred to as the number of particles at x. Moreover if Λ ⊂ Λ η Λ will denote the restriction of η to Λ . For two elements σ, ξ ∈ Ω Λ , the operations σ ± ξ are defined componentwise (for the difference whenever it returns an element of Ω Λ ). In what follows, given x ∈ Λ, we make use of the special configuration δ x , having one particle at x and no other particle. For f : Ω Λ → R and x, y ∈ Λ, we let Consider, at a formal level, the operator where y ∼ x means |x − y| = 1, and c : N → R + is a function such that c(0) = 0 and inf{c(n) : n > 0} > 0. In the case of Λ finite, for each N ∈ N \ {0}, L Λ is the infinitesimal generator of a irreducible Markov chain on the finite state space {η ∈ Ω Λ : η Λ = N }, where is the total number of particles in Λ. The unique stationary measure for this Markov chain is denoted by ν N Λ and is given by where c(0)! := 1, c(k)! := c(1) · · · · · c(k), for k > 0, and Z N Λ is the normalization factor. The measure ν N Λ will be referred to as the canonical measure. Note that the system is reversible for ν N Λ , i.e. L Λ is self-adjoint in L 2 (ν N Λ ) or, equivalently, the detailed balance condition holds for every x ∈ Λ and η ∈ Ω Λ such that η x > 0.
Our main result, that is stated next, will be proved under the following conditions.
As remarked in [5] for the spectral gap, N -independence of the logarithmic-Sobolev constant requires extra-conditions; in particular, our main result would not hold true in the case The following condition, that is the same assumed in [5], is a monotonicity requirement on c(·) that rules out the case above.

Condition 2.2 (M)
There exists k 0 > 0 and a 2 > 0 such that c(k) − c(j) ≥ a 2 for any j ∈ N and k ≥ j + k 0 .
A simple but key consequence of conditions above is that there exists A 0 > 0 such that In what follows, we choose Λ = [0, L] d ∩ Z d . In order to state our main result, we define the Dirichlet form corresponding to L Λ and ν N Λ : (2.5) Theorem 2.1 Assume that conditions (LG) and (M) hold. Then there exists a constant C(L) > 0, that may only depend on a 1 , a 2 , the dimension d and L, such that for every choice of N ≥ 1, L ≥ 2 and f : Ω Λ → R, f > 0, we have

Outline of the proof
For simplicity, the proof will be outlined in one dimension. The essential steps for the extension to any dimension are analogous to the ones for the spectral gap, that can be found in [4], Appendix 3.3. However, in the most technical and original estimates contained in this work (see Sections 7 and 8), proofs are given in a general dimension d ≥ 1.

Step 1: duplication
The idea is to prove Theorem 2.1 by induction on |Λ|. Suppose |Λ| = 2L, so that Λ = Λ 1 ∪Λ 2 , |Λ 1 | = |Λ 2 | = L, where Λ 1 , Λ 2 are two disjoint adjacent segments in Z. By a basic identity on the entropy, we have . Thus, by the tensor property of the entropy (see [1], Th. 3.2.2): Now, let s(L, N ) be the maximum of the logarithmic-Sobolev constant for the zero-range process in volumes Λ with |Λ| ≤ L and less that N particles, i.e. s(L, N ) is the smallest constant such that . for all f > 0, |Λ| ≤ L and n ≤ N . Then, by (3.2), Identity (3.1) and inequality (3.3) suggest to estimate s(L, N ) by induction on L. The hardest thing is to make appropriate estimates on the term Ent ν N Λ (ν N Λ [f |η Λ 1 ]). Note that this term is the entropy of a function depending only on the number of particles in Λ 1 .

3.2
Step 2: logarithmic Sobolev inequality for the distribution of the number of particles in Λ 1 is a probability measure on {0, 1, . . . , N } that is reversible for the birth and death process with generator and Dirichlet form Logarithmic Sobolev inequalities for birth and death processes are studied in [8]. The nontrivial proof that conditions in [8] are satisfied by γ(n), leads to the following result.
We now apply Proposition 3.1 to the second summand of the r.h.s. of (3.1), and we obtain One of the key points in the proof of Theorem 2.1 consists in finding the "right" representation for the discrete gradient , that appears in the r.h.s. of (3.5).
Moreover, by exchanging the roles of Λ 1 and Λ 2 , the r.h.s. of (3.6) can be equivalently written as, for every n − 0, 1, . . . , N − 1, The representations (3.6) and (3.7) will be used for n ≥ N 2 and n < N 2 respectively. For convenience, we rewrite (3.6) and (3.7) as and (3.10) Thus, our next aim is to get estimates on the two terms in the r.h.s. of (3.8). It is useful to stress that the two terms are qualitatively different. Estimates on A(n) are essentially insensitive to the precise form of c(·). Indeed, the dependence of A(n) on L and N is of the same order as in the case c(n) = λn, i.e. the case of independent particles. Quite differently, the term B(n) vanishes in the case of independent particles, since, in that case, the term x∈Λ 1 y∈Λ 2 h(η x )c(η y ) is a.s. constant with respect to ν N Λ ·|η Λ 1 = n − 1 . Thus, B(n) somewhat depends on interaction between particles. Note that our model is not necessarily a "small perturbation" of a system of independent particles; there is no small parameter in the model that guarantees that B(n) is small enough. Essentially all technical results of this paper are concerned with estimating B(n).

Step 4: estimates on A(n)
The following proposition gives the key estimate on A(n).

Proposition 3.3 There is a constant C > 0 such that
Remark 3.4 Let us try so see where we are now. Let us ignore, for the moment the term B(n), i.e. let us pretend that B(n) ≡ 0. Thus, by (3.8) and Proposition 3.3 we would have Inserting (3.11) into (3.5) we get, for some possibly different constant C, where we have used the obvious identity: for some C > 0, i.e. we get the exact order of growth of s(L, N ). In all this, however, we have totally ignored the contribution of B(n).

Step 5: preliminary analysis of B(n)
We confine ourselves to the analysis of B(n) for n ≥ N 2 , since the case n < N 2 is identical. Consider the covariance that appears in the r.h.s. of the first formula of (3.10). By elementary properties of the covariance and the fact that It follows by Conditions (LG) and (M) (see (2.4)) that, for some constant C > 0, h(η x ) ≤ C and c(η y ) ≤ Cη y . Thus, a simple estimate on the two summands in (3.16), yields, for some Thus, our next aim is to estimate the two covariances in (3.17).
3.6 Estimates on B(n): entropy inequality and estimates on moment generating functions By (3.17), estimating B 2 (n) consists in estimating two covariances. In general, covariances can be estimated by the following entropy inequality, that holds for every probability measure ν (see [1], Section 1.

2.2)
: where f ≥ 0 and t > 0 is arbitrary. Since in (3.17) we need to estimate the square of a covariance, we write (3.18) with −g in place of g, and obtain Therefore, we first get estimates on the moment generating functions ν e ±t(g−ν[g]) , and then optimize (3.19) over t > 0. Note that the covariances in (3.17) involve functions of η Λ 1 or η Λ 2 . In next two propositions we write Λ for Λ 1 and Λ 2 , and denote by N the number of particles in Λ. Their proof can be found in [3].
Proposition 3.6 There exists a constant C L depending on L = |Λ| such that the following inequalities hold: Inserting these new estimates in (3.17) we obtain, for some possibly different C L , (3.24) In order to simplify (3.24), we use Proposition 7.5, which gives for some C > 0. It follows that and Thus, (3.24) implies, for some C L > 0 depending on L, recalling also that n ≥ N 2 , Now, we bound the two terms and insert these estimates in (3.6). What comes out is then used to estimate (3.5), after having obtained the corresponding estimates for n < N 2 . Recalling the estimates for A(n), straightforward computations yield To deal with the term ν N Λ [f ] in (3.28) we use the following well known argument. Set By Rothaus inequality (see [1], Lemma 4.3.8) Using this inequality and replacing f by f in (3.28) we get, for a different C L , where, in the last line, we have used the Poincaré inequality (1.3). Therefore that implies (2.6), provided we prove the following "basis step" for the induction.
The proof of Proposition 3.7 is also given in Section 8. As we pointed out above, (3.30) gives no indication on how s(L) = sup N s(N, L) grows with L. The proof of the actual L 2 -growth is given in [3].
Proof. We begin by writing and similarly Thus Letting f ≡ 1 in this last formula we obtain and, therefore, Proof of Proposition 3.2. Equation (3.6) is obtained by (4.1) by averaging over y ∈ Λ 2 .

Bounds on A(n): proof of Proposition 3.3
Proof of Proposition 3.3. The Proposition is proved if we can show that for 1 ≤ n < N/2 We will prove only this last bound being the proof of the previous one identical. So assume n ≥ N/2 and notice that where a := h +∞ ((2.4) implies boundedness of h). In order to bound the last factor in (5.2) we observe that by Lemma 4.1 where the fact c(k) ≤ ak (see (2.4)) has been used to obtain the last line. This implies that where in last step we used Proposition 7.5. By plugging this bound in (5.2) we get Now we have to bound the last factor in the right hand side of (5.3). For x, y ∈ Z the path between x and y will be the sequence of nearest-neighbor integers γ(x, y) By Jensen inequality, we obtain By plugging this bound in (5.3) we get (5.1).
6 Rough estimates: proofs of Propositions 3.5 and 3.6 Proof of Proposition 3.5. We begin with the proof of (3.20). Denote by Ω N Λ the set of configurations having N particles in Λ. Then where Z N Λ is a normalization factor. It is therefore easily checked that for any x ∈ Λ. Thus, for every function f , where , and so we rewrite (6.2) as Now, let t > 0, and define ϕ(t) = ν N Λ e tc(ηx) .
We now estimate ϕ(t) by means of the so-called Herbst argument (see [1], Section 7.4.1). By direct computation, Jensen's inequality and (6.3), we have We now claim that, for every x ∈ Λ and 0 ≤ t ≤ 1 for some constant C L depending on L but not on N . For the moment, let us accept (6.5), and show how it is used to complete the proof. By (6.4), (6.5) and the fact that for 0 ≤ t ≤ 1 and some possibly different C L . Equivalently, letting , by (6.6) and Gronwall lemma we have from which (3.20) easily follows, for t ≥ 0. We now prove (6.5). We shall use repeatedly the inequality which holds for x, y ∈ R. Using (6.7) and the fact that, by condition (LG), from which we have that (6.5) follows if we show At this point we use the notion of stochastic order between probability measures. For two probabilities µ and ν on a partially ordered space X, we say that ν ≺ µ if f dν ≤ f dµ for every integrable, increasing f . This is equivalent to the existence of a monotone coupling of ν and µ, i.e. a probability P on X × X supported on {(x, y) ∈ X × X : x ≤ y}, having marginals ν and µ respectively (see e.g. [6], Chapter 2). As we will see, (6.8) would not be hard to prove if we had ν N −1 Under assumptions (LG) and (M) a slightly weaker fact holds, namely that there is a constant B > 0 independent of N and L such that if N ≥ N + BL then ν N Λ ≺ ν N Λ (see [5], Lemma 4.4). In what follows, we may assume that N > BL. Indeed, in the case N ≤ BL there is no real dependence on N , and (3.20) can be proved by observing that Denote by Q the probability measure on N Λ × N Λ that realizes a monotone coupling of ν N −1 as marginals, and Using again (6.7) Thus, it follows that, for some C > 0, Inserting this inequality in (6.11), we get, for t ≤ 1 and some C L > 0, With the same arguments, it is shown that In order to put together (6.12) and (6.13), we need to show that, for 0 ≤ t ≤ 1, for some L-dependent C L . By condition (LG) and (6.3) we have where, for the last step, we have used the facts that, for some > 0, s. (for both inequalities we use (2.4)). The proof (3.20) is now completed for the case t ≥ 0. For t < 0, it is enough to observe that our argument is insensitive to replacing c(·) with −c(·).
We now prove (3.21) The idea is to use the fact that the tails of that, together with the previously obtained identity where, in the last step, we use the fact that ν N +1 Λ [c(η x )] ≥ N/L for some > 0. In (6.18) and in what follows, the L-dependence of constants is omitted. For ρ > 0, let c(ρ) be obtained by linear interpolation of c(n), n ∈ N. Observe that, by (1. for some C > 0, possibly depending on L. From (6.18) and (6.19) it follows that, for some Moreover, from Condition (M) it follows that there is a constant C > 0 such that Thus, for every M > 0 there exists C > 0 such that It follows that, for all M > 0 Note that (6.22) is trivially true for M ≤ 0, so it holds for all M ∈ R. Now, take t ∈ (0, 1]. We have where, in the last step, we have used (3.20). This completes the proof for the case t ∈ (0, 1]. For t ∈ [−1, 0) we proceed similarly, after having replaced, in (6.22 Proof of Proposition 3.6. We first prove inequality (3.22). Clearly, it is enough to show that, for all x ∈ Λ, By the entropy inequality (3.19) and (3.20) we have, for 0 < t ≤ 1: .

Local limit theorems
The rest of the paper is devoted to the proofs of all technical results that have been used in previous sections. For the sake of generality, we state and prove all results in dimension d ≥ 1.
Using the language of statistical mechanics, having defined the canonical measure ν N Λ , for ρ > 0 we consider the corresponding grand canonical measure where α(ρ) is chosen so that µ ρ (η x ) = ρ, x ∈ Λ, and Z(α(ρ)) is the corresponding normalization. Clearly µ ρ is a product measure with marginals given by (1.2). Monotonicity and the Inverse Function Theorem for analytic functions, guarantee that α(ρ) is well defined and it is a analytic function of ρ ∈ [0, +∞). We state here without proof some direct consequences of Conditions 2.1 and 2.2. The proofs of some of them can be found in [5].

Local limit theorems for the grand canonical measure
The next two results are a form of local limit theorems for the density of η Λ under µ ρ . Define p ρ Λ (n) := µ ρ [η Λ = n] for ρ > 0, n ∈ N and Λ ⊂⊂ Z d . The idea is to get a Poisson approximation of p N/|Λ| Λ (n) for very small values of N/|Λ| and to use the uniform local limit theorem (see Theorem 6.1 in [5]) for the other cases.
Proof. Let ρ := N/|Λ| and assume, without loss of generality, that 0 ∈ Λ. Notice that We begin by proving that Since which establishes (7.4). Now let A trivial calculation shows that and finally Observe that for any n ∈ {0, . . . , |Λ|}, we have This comes from the fact that the random variables {η x : x ∈ Λ}, under the probability measure µ ρ [·| max x∈Λ η x ≤ 1], are Bernoulli independent random variables with mean ρ. The remaining part of the proof follows the classical argument of approximation of the binomial distribution with the Poisson distribution. Using (7.5) and (7.6), after some simple calculations we get uniformly in 0 < N ≤ N 0 and 0 ≤ n < N . This proves that there exist positive constants v 1 and B 3 such that if Λ ⊂⊂ Z d is such that |Λ| > v 1 then with N ≤ N 0 and n ∈ N with n ≤ N . The general case follows easily because the set of n ∈ N, N ∈ N \ {0} and Λ ⊂⊂ Z d such that n ≤ N ≤ N 0 , |Λ| ≤ v 1 and 0 ∈ Λ is finite.
Proof. This is a special case of the Local Limit Theorem for µ ρ (see Theorem 6.1 in [5]).
We conclude this section with a bound on the tail of p ρ Λ which will be used in the regimes not covered by Lemma 7.2 or Proposition 7.3.

Lemma 7.4
There exists a positive constant A 0 such that Proof. Notice that and, by the change of variable ξ := σ − δ x : This means that By (2.4) we know that there exists a positive constant B 0 such that Thus, by plugging these bounds in (7.9), we get from which the conclusion follows.

Gaussian estimates for the canonical measure
In this section we will prove some Gaussian bounds on ν N Λ [η Λ = ·], when the volumes |Λ| and |Λ | are of the same order (typically it will be |Λ |/|Λ| = 1/2). These bounds are volume dependent (see Proposition 7.8 below) and so are of limited utility. However they will be used to prove Proposition 7.9 which, in turn, is used in Section 8 to regularize ν N Λ [η Λ = ·] and prove Gaussian uniform estimates on it.
In this case we define .

(7.10)
We begin by proving a simple result on the decay of the tails of γ N Λ .
Next we prove Gaussian bounds on γ N Λ . This is a very technical argument. We begin with the case |Λ| = 2.
Lemma 7.6 Assume that |Λ| = 2, and defineN := N/2 for N ∈ N \ {0}. There exist a positive constant A 0 such that Proof. We split the proof in several steps for clarity purpose.
Step 2. There exists where we used the fact that n ≤ 3N/4; thus Since log(1 − x) ≥ −2x, for x ∈ [0, 1/2], then By plugging this bound into (7.21) we obtain which completes the proof of (7.20) in the case N ≥ n 1 . The general case is obtained by a a finiteness argument.
(7.27) By (7.24) there exists B 1 > 0 such that In order to bound the product factor in the right hand side of (7.27), notice that by Proposition 7.5 there exists B 2 > 0 such that for any k ∈ {1, . . . , N − 1}, thus By plugging this bound and (7.28) into (7.27), we get (7.26).
Step 6. There exists A 6 > 0 such that if N ∈ N \ {0}, then, Proof of Step 6. By (7.22), (7.24) and (7.26) we obtain B 1 > 0 such that for any for any n ∈ {0, . . . , N }. By summing for n ∈ {0, . . . , N } we have Thus we are done if we can show that, for any B 2 > 0, there exists B 3 > 0 such that √N that proves the upper bound in (7.30). The proof of the lower bound is similar.
We now prove an iterative procedure that allow us to extend the Gaussian bounds of Lemma 7.6, from |Λ| = 2 to a generic Λ ⊂⊂ Z d .

Further estimates on the grand canonical measure
The next result is a uniform estimate on µ ρ [η Λ = ρ|Λ|], see (7.42) below, and will be used in Section 8. If c(k) = k the result can be obtained elementarily from the Stirling Formula. The general case is more difficult.
Very small density case. For any fixed N 0 ∈ N \ {0} Proof of very small density case. By Lemma 7.2 there exists B 2 > 0 such that for any Λ ⊂⊂ Z d Now taking v 1 large enough it easy to show that The general case follows again by a finiteness argument.
Normal density case. There exist v 0 > 0 such that Proof of normal density case. By point 2. of Proposition 7.3 there exist positive constants B 3 and v 4 such that: uniformly in N ∈ N \ {0} and Λ ⊂⊂ Z d such that N/|Λ| > ρ 0 . Now it is easy to show that there exists v 0 > 0 such that for any Λ ⊂⊂ Z d with |Λ| > v 0 and any N ∈ N \ {0} such that N > ρ 0 |Λ| 2π .
8 One dimensional L.S.I.: proof of Proposition 3.1 In this section we prove Proposition 3.1, a logarithmic Sobolev inequality for a particular one dimensional birth and death process. Furthermore we will see that Proposition 3.1 implies Proposition 3.7.

A general result
Next result is Proposition A.5. in [2]. We report it only for completeness. Let {γ N : N ∈ N \ {0}} be a family of positive probability on Z. Assume that for any N ∈ N \ {0} the probability γ N is supported on {0, . . . , N }. It is elementary to check that γ N is reversible with respect to the continuous time birth and death process with rates for any N ∈ N \ {0}. The Dirichlet form of this Markov process may be written as Proof. The proof follows closely the proof of Proposition A.5 in [2]. By Proposition A.1 of [2] we have to bound from above B 0 (N ) : By plugging this bound into (8.6) we obtain (8.4). Using the same argument and (8.2) we obtain (8.5).
Assume now that n ∈ {N − √N + 1, . . . , N }. Then, again by condition (8.3), by taking N ≥ n 0 and n 0 large enough. By plugging this bound into (8.11) we get (8.9) in this case also.
We can now conclude the proof of the proposition. We will bound from above By the previous bound and (8.12) we have that B + 0 (N ) ≤ A 1 N for any N ∈ N \ {0}.

Proof of Proposition 3.1
Recall that for any Λ ⊂ Λ ⊂⊂ Z d such that |Λ| = 2|Λ | and any N ∈ N \ {0} we defined γ N Λ = ν N Λ (η Λ = ·). We have to show that any f : The assumptions of Proposition 8.1 are too strong to be fulfilled by γ N = γ N Λ . In particular, while assumption (8.3) holds, as we will see a fortiori, assumptions (8.1) and (8.2) may not be fulfilled in general (the case |Λ| = 2 may be instructive to see this). So following [2], for any ∈ (0, 1/4), we consider a regularization γ N, and γ N, It easy to check that γ N, Λ is a probability density supported on {0, . . . , N }. Now we show that γ N, Lemma 8.2 For any fixed ∈ (0, 1/4) there exists a positive constant A 0 such that Proof. Fix ∈ (0, 1/4), since γ N Λ (n)/ γ N, Λ (n) = 1 for any n ∈ I N, , we have to bound the ratio γ N Λ (n)/ γ N, Λ (n) for n ∈ I N, only. So assume that n ∈ I N, and define . Therefore min n,m∈I N, , for any n ∈ I N, , N ∈ N \ {0} and Λ ⊂⊂ Z d with |Λ| even. Furthermore notice that This formula shows that it is possible to extend the function H N Λ : {0, . . . , N } → R to a real function, H N Λ : [0, N ] → R by defining for any x ∈ [0, N ]. A direct calculation gives Since which, by equation (1.3) of [5] (namely α (ρ) = α(ρ)σ 2 (ρ)), implies Differentiating this identity and using again equation (1.3) of [5], we obtain which, again with (7.2), implies that there exist a positive constant B 1 such that for any Assume now that x ∈ [N , (1 − )N ]. Then the previous inequality yields it follows that, if > 0 is small enough, there exists a positive constant B 2 ( ) such that Integrating this inequality with respect to x fromN to y ∈ [N , (1 − )N ], we obtain Now take n ∈ I N, such that n >N and integrate the previous inequality with respect to y from n to n + 1 to obtain Equation ( Proof. We split the proof in several steps for clarity purpose.
Step 2. There exists A 2 > 0 such that if N ∈ N \ {0} then and becauseN ≤ N , In order to bound the product factor in the right hand side of (8.27), notice that by Proposition 7.5 there exists B 2 > 0 such that for any k ∈ {1, . . . , N − 1}. Thus  We claim that there exists a constant B 0 > 0 such that for any ϕ : N → R and any N ∈ N \ {0} we have N D(ϕ, ϕ) ≤ B 0 D(ϕ, ϕ). (8.31) In this case we will have by Proposition 3.1 which is (2.6) in the present case. Thus we have to verify (8.31). Observe that Since, by (7.10), γ N Λ (n)/γ N Λ (n − 1) = c(N − n + 1)/c(n) we get D(ϕ, ϕ) =