Annealed Potts models on rank-1 inhomogeneous random graphs

TL;DR

This paper analyzes the phase transition behavior of the annealed Potts model on rank-1 inhomogeneous random graphs, revealing conditions under which the transition is first or second order, especially for Pareto-distributed weights.

Contribution

It establishes the existence of the thermodynamic limit for general weights, characterizes phase transition orders for various distributions, and identifies critical parameters for Pareto weights.

Findings

Thermodynamic limit exists for general vertex weights.

Phase transition is first order for finite-variance weights with certain conditions.

Transition order depends on Pareto tail exponent, with explicit thresholds.

Abstract

In this paper, we study the annealed ferromagnetic $q$-state Potts model on sparse rank-1 random graphs, where vertices are equipped with a vertex weight, and the probability of an edge is proportional to the product of the vertex weights. In an annealed system, we take the average on both numerator and denominator of the ratio defining the Boltzmann-Gibbs measure of the Potts model. We show that the thermodynamic limit of the pressure per particle exists for rather general vertex weights. In the infinite-variance weight case, we show that the critical temperature equals infinity. For finite-variance weights, we show that, under a rather general condition, the phase transition is {\em first order} for all $q\geq 3$. However, we cannot generally show that the discontinuity of the order parameter is {\em unique}. We prove this uniqueness under a reasonable condition that holds for various distributions, including uniform, gamma, log-normal, Rayleigh and Pareto distributions. Further, we show that the first-order phase transition {\em persists} even for some small positive external field. In the rather relevant case of Pareto distributions with power-law exponent $\tau$, remarkably, the phase transition is first order when $\tau\geq 4$, but not necessarily when the weights have an infinite third-moment, i.e., when $\tau\in(3,4)$. More precisely, the phase transition is second order for $\tau\in (3,\tau(q)]$, while it is first order when $\tau>\tau(q)$, where we give an explicit equation that $\tau(q)$ solves.