Probability Seminar

Moumanti PodderGeorgia Institute of Technology
Sofic and percolative entropies of Gibbs measures on regular infinite trees

Monday, March 19, 2018 - 4:00pm
Malott 406

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the percolative entropy $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. When it is known to exist, the limit of $|V_n|^{-1}H(\mu_n)$ is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly.