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

Abstract

Abstract 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

$$\alpha _2$$

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,

$$p\ge 1$$

p ≥ 1

). As an example, we show

$$\alpha _1 = 1+\frac{1+o(1)}{\log q}$$

α 1

= 1 +

1 + o ( 1 )

log q

. Furthermore, p-NLSIs allow us to conclude that for

$$q\ge 3$$

q ≥ 3

, distributions that are not a product of identical distributions can have slower speed of convergence to equilibrium, unlike the case

$$q=2$$

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

$$\begin{aligned} \frac{\log q}{\log q - \log (q-1)} = (1-o(1))q\log q. \end{aligned}$$

log q

log q - log ( q - 1 )

=

( 1 - o

( 1 )

)

q log 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

$$q\ge 3$$

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.