Limits of discrete distributions and Gibbs measures on random graphs

TL;DR

This paper develops a framework connecting graph limits, Gibbs measures, and regularity lemmas to analyze the asymptotic behavior of distributions on sparse random graphs, confirming predictions from physics.

Contribution

It introduces continuous embeddings of discrete distributions using graph limit theory and applies these to Gibbs measures on sparse graphs, verifying the replica symmetric solution.

Findings

Established a notion of convergence for distributions via graph limits

Derived a version of Szemerédi's regularity lemma for these distributions

Confirmed the replica symmetric solution in the non-reconstruction regime

Abstract

Building upon the theory of graph limits and the Aldous-Hoover representation and inspired by Panchenko's work on asymptotic Gibbs measures (Annals of Probability 2013), we construct continuous embeddings of discrete probability distributions. We show that the theory of graph limits induces a meaningful notion of convergence and derive a corresponding version of the Szemer\'edi regularity lemma. Moreover, complementing recent work (Bapst et. al. 2015), we apply these results to Gibbs measures induced by sparse random factor graphs and verify the "replica symmetric solution" predicted in the physics literature under the assumption of non-reconstruction.