Glauber versus Kawasaki for Spectral Gap and Logarithmic Sobolev Inequalities of Some Unbounded Conservative Spin Systems

#### D. Chafai

2003, v.9, Issue 3, 341-362

ABSTRACT

Inspired by the recent results by Landim, Panizo and Yau [C. Landim, G. Panizo and H. T. Yau, Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems, Ann. Inst. H. Poincare, Probabilites et Statistiques, 2002, 38, N5, 739-777] 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 $$\sum_{i=1}^n V(x_i)+V\big(\MSX\big), \quad (x_1,\ldots,x_n)\in\r^n.$$ Specifically, we examine the case $V$ is strictly convex (or small perturbation of strictly convex) and, following the mentioned paper by Landim, Panizo and Yau, 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 yield, 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 the mentioned paper by Landim, Panizo and Yau, relying in particular on the Lu - Yau martingale decomposition and clever partitionings of the conditional measure.

Keywords: interacting particle systems,spectral gap,Poincare inequality,log-Sobolev inequality,conservative spin systems,continuous spin systems,Ginzburg - Landau process on a lattice,Glauber dynamics,Kawasaki dynamics,mean-field models,exchangeable measures