A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees

TL;DR

This paper introduces a new entropy-based criterion for determining when a Markov chain on a Galton-Watson tree is non-reconstructible, extending previous results and providing improved bounds especially for asymmetric models.

Contribution

It develops a symmetric entropy bound criterion for non-reconstruction in Markov chains on Galton-Watson trees, applicable to non-reversible matrices and simplifying proofs for known models.

Findings

Reproduces earlier results for symmetric Ising models

Provides improved numerical bounds for asymmetric Ising models

Establishes a general recursion formula for symmetrized relative entropy

Abstract

We give a criterion of the form Q(d)c(M)<1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of M's with a given invariant distribution, that is defined in terms of a (q-1)-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition-Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible M and its proof is based on a general Recursion Formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.