 0230 Chafai D.
 Glauber versus Kawasaki for spectral gap and logarithmic Sobolev
inequalities of some unbounded conservative spin systems
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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
$$
\sum_{i=1}^n V(x_i)+V(Mx_1\cdotsx_n), \quad (x_1,\ldots,x_n)\in R^n
$$
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 LuYau martingale decomposition and
clever partitionings of the conditional measure.
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