Poincaré and logarithmic Sobolev inequality for Ginzburg-landau processes in random environment

Abstract

We consider reversible, conservative Ginzburg-Landau processes in a random environment, whose potential are bounded perturbations of the Gaussian potential, evolving on a d-dimensional cube of length L. We prove in all dimensions that the spectral gap of the generator and the logarithmic Sobolev constant are of order L −2 almost surely with respect to the environment.

Convergence to equilibrium of conservative particle systems on $\mathbb{Z}^{\bm{d}}$

“…In the context of the present article this result has been proved by [8] under assumptions (H2) using the martingale method introduced by [12]. It is only for this reason that we impose (H2).…”

“…Expectation with respect to ν ,M is denoted by E ,M . From [8] we have the following spectral gap estimate. …”

“…In particular, the first variance of (4.7) is bounded by By formula (3.10) in [8], the second variance in (4.7) is bounded above by…”

“…In particular, by Corollary 6.4 in [8] and the logarithmic Sobolev inequality for Glauber dynamics (Lemma 4.1 in [8]), there exists a finite constant C 0 such that…”

“…By Corollary 6.3 in [8], by the logarithmic Sobolev inequality for the GinzburgLandau process (Theorem 2.2 in [8]) and by the Schwarz inequality, the second term in (5.7) is bounded above by…”