Right-convergence of sparse random graphs

TL;DR

This paper proves the right-convergence of sparse Erdős-Rényi graphs for certain target graphs W, addressing fundamental difficulties in sparse graph limits and confirming a special case of Talagrand's conjecture using interpolation techniques.

Contribution

It establishes the right-convergence of sparse random graphs for target graphs W with a convexity property, extending previous results and confirming a case of Talagrand's conjecture.

Findings

Proves right-convergence for sparse Erdős-Rényi graphs with convexity conditions.

Addresses the existence of free energy limits in sparse graph models.

Confirms a special case of Talagrand's conjecture regarding measure limits.

Abstract

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs $\G_N, N\ge 1$ to some target graph $W$. The theory of dense graph convergence, including random dense graphs, is now well understood, but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the log-partition function limits, also known as free energy limits, appropriately normalized for the Gibbs distribution associated with $W$. In this paper we prove that the sequence of sparse \ER graphs is right-converging when the tensor product associated with the target graph $W$ satisfies certain convexity property. We treat the case of discrete and continuous target graphs $W$. The latter case allows us to prove a special case of Talagrand's recent conjecture (more accurately stated as level III Research Problem 6.7.2 in his recent book), concerning the existence of the limit of the measure of a set obtained from $\R^N$ by intersecting it with linearly in $N$ many subsets, generated according to some common probability law. Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli and developed further in a series of papers. Specifically, Bayati et al establish the right-convergence property for Erdos-Renyi graphs for some special cases of $W$. In this paper most of the results in this paper follow as a special case of our main theorem.