Decay of Correlations for the Hardcore Model on the $d$-regular Random Graph

TL;DR

This paper analyzes the decay of correlations in the hardcore model on random regular graphs, showing convergence to tree Gibbs measures and equating reconstruction thresholds with those on regular trees.

Contribution

It establishes the local weak limit of the hardcore model on random regular graphs below the condensation threshold and links the reconstruction thresholds to those on the infinite regular tree.

Findings

Convergence of local measures to tree Gibbs measures

Reconstruction threshold on graphs equals that on trees

Asymptotic determination of the reconstruction threshold

Abstract

A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs asymptotically until just below its condensation threshold, showing that it converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the $d$-regular tree for which we determine precise asymptotics. We expect that our methods will generalize to a wide range of spin systems for which the second moment method holds.