02-30 Chafai D.

Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems (61K, LaTeX 2e) Jan 21, 02

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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 $$ \sum_{i=1}^n V(x_i)+V(M-x_1\cdots-x_n), \quad (x_1,\ldots,x_n)\in R^n $$ Specifically, we examine the case $V$ is strictly convex (or small perturbation of strictly convex) and, following [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.

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