Non-linear Log-Sobolev Inequalities for the Potts Semigroup and Applications to Reconstruction Problems

Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. Diaconis and Saloff-Coste (Ann Appl Probab 6(3):695–750, 1996) computed the maximum ratio between the relative entropy and the Dirichlet form, obtaining the constant α _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≥ 1 ). As an example, we show α _1 = 1+1+o(1)/log q . Furthermore, p -NLSIs allow us to conclude that for q≥ 3 , distributions that are not a product of identical distributions can have slower speed of convergence to equilibrium, unlike the case q=2 . By integrating the 1-NLSI we obtain new strong data processing inequalities (SDPI), which in turn allows us to improve results of Mossel and Peres (Ann Appl Probab 13(3):817–844, 2003) on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a q -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 log q/log q - log (q-1) = (1-o(1))qlog q. This recovers previous results (of Sly in Commun Math Phys 288(3):943–961, 2009 and Bhatnagar et al. in SIAM J Discrete Math 25(2):809–826, 2011) in (slightly) more generality, but more importantly avoids the need for any coloring-specific arguments. Similarly, we improve the state-of-the-art on the weak recovery threshold for the stochastic block model with q balanced groups, for all q≥ 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. (Combin Probab Comput 31(6):1048–1069, 2022). These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics.