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

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.