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Presented By: Department of Mathematics

Financial/Actuarial Mathematics Seminar

Sofic and percolative entropies of Gibbs measures on regular infinite trees

Consider a statistical physical model on the infinite $d$-regular tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite $d$-regular graphs with vertex sets $V_{n}$ that locally converge to $T_{d}$. From $\Phi$, one can construct a sequence of corresponding Gibbs measures $\left\{\mu_{n}\right\}_{n \in \mathbb{N}}$ on the graphs $G_n$. 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 \emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ when $\Phi$ 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.
Joint work with Tim Austin, UCLA Mathematics Speaker(s): Moumanti Podder (University of Washington)

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