Reconstruction on trees and spin glass transition

TL;DR

This paper links the problem of reconstructing information on noisy tree networks to the glass transition in statistical physics, deriving bounds and applying numerical methods to understand phase transitions and coloring problems.

Contribution

It establishes a correspondence between reconstruction thresholds and glass transition points, providing new bounds and insights into graph coloring and spin glass models.

Findings

Reconstruction threshold coincides with the glass transition point.

Derived new rigorous bounds on the reconstruction problem.

Applied numerical methods to predict thresholds in various channels.

Abstract

Consider an information source generating a symbol at the root of a tree network whose links correspond to noisy communication channels, and broadcasting it through the network. We study the problem of reconstructing the transmitted symbol from the information received at the leaves. In the large system limit, reconstruction is possible when the channel noise is smaller than a threshold. We show that this threshold coincides with the dynamical (replica symmetry breaking) glass transition for an associated statistical physics problem. Motivated by this correspondence, we derive a variational principle which implies new rigorous bounds on the reconstruction threshold. Finally, we apply a standard numerical procedure used in statistical physics, to predict the reconstruction thresholds in various channels. In particular, we prove a bound on the reconstruction problem for the antiferromagnetic ``Potts'' channels, which implies, in the noiseless limit, new results on random proper colorings of infinite regular trees. This relation to the reconstruction problem also offers interesting perspective for putting on a clean mathematical basis the theory of glasses on random graphs.