Reconstruction for the Asymmetric Ising Model on Regular Trees

TL;DR

This paper refines the analysis of the asymmetric Ising model on regular trees to establish the precise conditions under which the Kesten-Stigum bound accurately predicts the reconstruction threshold, extending previous symmetric results.

Contribution

It introduces a refined moment recursion analysis for the asymmetric Ising model, identifying the critical condition where the Kesten-Stigum bound is tight for reconstruction.

Findings

Kesten-Stigum bound is tight for the asymmetric Ising model on regular trees.

Refined analysis extends symmetric results to asymmetric cases.

Critical condition for reconstruction threshold established.

Abstract

It is known that the Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels. In this paper, we will adopt a refined analysis of moment recursion on a weighted version of the magnetization, which is engaged in Sly [2011] to handle the symmetric Potts model, and establish the critical condition of the asymmetric Ising model to make Kesten-Stigum bound the reconstruction threshold on regular $d$-ary trees.