g$, we also write $f \simeq g$. Finally we denote by ${{\mathbb N}}=\{1,2,\dots\}$ the set of positive integers and by ${{\mathbb N}}_0=\{0\}\cup{{\mathbb N}}$ the set of non-negative integers.

Invariant measures and Dirichlet form
-------------------------------------

For $\lambda \in (0,1)$, let $\nu^n_\lambda$ be the *Bernoulli product measure* on $\Omega_n$ with density $\lambda$: its marginals satisfy $$\nu^n_\lambda\big\{\eta: \eta(x)=1\big\} = 1-\nu^n_\lambda\big\{\eta:\eta(x)=0\big\} =\lambda, \quad  \textrm {for each } x\in \Lambda_n.$$ When $\lambda=\rho$, the measure $\nu^n_\rho$ is invariant but not reversible with respect to the evolution of $\{\eta_t^n\,; \,t {\geqslant}0\}$. To ensure the invariance, it is enough to check that $
\int \mathcal L_n f(\eta)\, \nu^n_\rho(d\eta) =0\,,$ for any function $f:\Omega_n\to {{\mathbb R}}$. This is a long albeit elementary computation, which is omitted here.

In what follows we consider that the initial measure of the Markov process $\{\eta^n_{tn^2}\,; \, t\in [0,T]\}$ is the invariant measure $\nu^n_\rho$. For short, we denote the probability measure ${{\mathbb P}}_{\nu^n_\rho}$ by ${{\mathbb P}}_\rho$ and the corresponding expectation by ${{\mathbb E}}_\rho$.

The *Dirichlet form* ${{\mathfrak D}}_n$ is the functional acting on functions $f:\Omega_n\to {{\mathbb R}}$ which belong to $L^2(\nu^n_\rho)$ as: $$\label{eq:dir_form}
{{\mathfrak D}}_n(f) =\int f(\eta)\,(-{{\mathcal L}}_nf)(\eta)\, \nu^n_\rho(d\eta)\,.$$ Invoking a general result [@kl Appendix 1, Prop. 10.1] we can rewrite ${{\mathfrak D}}_n$ as 