Gibbs conditioning, atypical consensus and splitting Gibbs measures on random regular graphs

TL;DR

This paper studies the behavior of opinions on random regular graphs conditioned on atypical consensus, showing convergence to specific Ising measures and revealing phase transitions and Gibbs measure properties.

Contribution

It characterizes the limiting distributions of opinions conditioned on atypical consensus in random regular graphs, linking them to Ising models and phase transition phenomena.

Findings

Conditional limits are Ising measures on infinite trees.

Phase transition occurs at a critical consensus value.

Limiting distributions relate to Gibbs measures on trees.

Abstract

Given n independent Bernoulli(p) random variables X_i, i = 1, ..., n, representing the opinions of individuals connected by an underlying random k-regular graph G_n on {1, ..., n}, we show that when conditioned on an atypical empirical consensus, which is the normalized sum of X_i X_j over neighboring vertices i, j, the joint distribution of the random variables converges, as n goes to infinity, to an Ising measure on the infinite k-regular tree T^k with a specific external field that depends only on the bias parameter p, and a temperature that depends on both p and the atypical consensus value. In particular, we show that conditional on the empirical consensus being smaller (respy, larger) than typical, the limit is a translation-invariant splitting (TIS) antiferromagnetic (respy, ferromagnetic) Ising measure on T^k. Moreover, if the bias is zero, then there is a phase transition: when the consensus exceeds k/(k-1), the conditional limits could be either the plus or minus boundary condition Ising measures. Furthermore, when X_i, i = 1, ..., n, are i.i.d. on a finite space, we show that when conditioned on an atypical value of the scaled sum of h(X_i, X_j) over neighboring vertices i and j, for any symmetric edge potential h, the limiting joint distribution of {X_i} lies in the set of (possibly degenerate) TIS Gibbs measures on T^k. The proofs leverage a tractable form of the large deviation rate function for component empirical measures of random regular graphs with i.i.d. marks and Gibbs conditioning principles, and entail careful analyses of associated non-convex constrained optimization problems. As a by-product of our results, we also obtain an (asymptotic) analog of the maximum entropy principle for Gibbs measures on random regular graphs.