Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems

Abstract

Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant $\alpha_2$ in the $2$-log-Sobolev inequality ($2$-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., $p$-NLSI, $p\ge1$). As an example, we show $\alpha_1 = 1+\frac{1+o(1)}{\log k}$. By integrating the $1$-NLSI we obtain the new strong data processing inequality (SDPI), which in turn allows us to improve results of Mossel and Peres on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a $k$-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least $$\frac{\log k}{\log k - \log(k-1)} = (1-o(1))k\log k.$$ This recovers previous results (of Sly and Bhatnagar et al.) in (slightly) more generality, but more importantly avoids the need for any coloring-specialized arguments. Similarly, we improve the state-of-the-art on the weak recovery threshold for the stochastic block model with $k$ balanced groups, for all $k\ge 3$. To further show the power of our method, we prove optimal non-reconstruction results for a broadcasting on trees model with Gaussian kernels, closing a gap left open by Eldan et al. These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics.