96-205 H.T. Yau
Logarithmic Sobolev Inequality for Generalized Simple Exclusion Processes (270K, ps) May 16, 96
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Abstract. Let $\bar \mu$ be a probability measure on the set $\{0, 1, \cdots , R\}$ for some $R\in \NN$ . Let $\L_L$ be a cube of width $L$ in $\ZZ^d$. Denote by $\mu^{gc}_{\L_L}$ the (grand canonical ) product measure with $\bar \mu$ as the marginal measure; here the superscript indicates the grand canonical ensemble. The canonical ensembles, denoted by $\mu^c_{\L_L, n}$, are defined by conditioning $\mu^{gc}_{\L_L}$ given that the total number of particle is $n$. Consider the zero-range exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site. The rates are chosen such that, for every $n$ and $L$ fixed, the measure $\mu^c_{\L_L, n}$ is reversible. We prove the logarithmic Sobolev inequality in the sense that $\int f \log f d\mu^c_{\L_L, n} \le \hbox{const.} L^2 D(\sqrt f)$ for any probability density $f$ with respect to $\mu^c_{\L_L, n}$; here the constant is independent of $n$ or $L$ and $D$ denote the Dirichlet form of the dynamics. The dependence on $L$ is optimal.

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