Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems

TL;DR

This paper compares Glauber and Kawasaki dynamics in unbounded conservative spin systems, establishing uniform bounds for spectral gap and logarithmic Sobolev inequalities, and connecting decay rates between the two dynamics.

Contribution

It provides new uniform bounds for Glauber dynamics in unbounded spin systems and relates these bounds to Kawasaki dynamics decay rates, extending previous results.

Findings

Uniform bounds for Glauber dynamics established

Classical L^{-2} decay for Kawasaki dynamics derived

Analysis relies on conservative approach and Lu-Yau martingale decomposition

Abstract

Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following [LPY], the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Glauber dynamics yields, in a transparent way, the classical L^{-2} decay for the Kawasaki dynamics on d-dimensional cubes of length L. The arguments of proofs however closely follow and make heavy use of the conservative approach and estimates of [LPY], relying in particular on the Lu-Yau martingale decomposition and clever partitionings of the conditional measure.