Annealed Potts models on rank-1 inhomogeneous random graphs
Cristian Giardinà, Claudio Giberti, Remco van der Hofstad, Guido Janssen, Neeladri Maitra
TL;DR
We study the annealed ferromagnetic $q$-state Potts model on sparse rank-1 inhomogeneous random graphs with vertex weights. The main methodological advance is a reduction to an inhomogeneous Curie–Weiss model, yielding a variational formula for the thermodynamic limit of the pressure per particle and a precise description of the optimizer structure. For finite-variance weights, the paper proves a first-order phase transition for all $q\ge 3$ under a zero-crossing criterion, with uniqueness of the order parameter established under a mild condition and persistence of the transition under small external fields; infinite-variance weights yield infinite critical temperature. In the Pareto-weight regime, the results show a tail-dependent smoothing: the transition is first-order for $\tau\ge 4$ and second-order for certain $\tau\in(3,4)$ regimes, with explicit critical criteria and a computable $t_c$, highlighting a rich interplay between weight distribution tails and mean-field spin behavior.
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 $τ$, remarkably, the phase transition is first order when $τ\geq 4$, but not necessarily when the weights have an infinite third-moment, i.e., when $τ\in(3,4)$. More precisely, the phase transition is second order for $τ\in (3,τ(q)]$, while it is first order when $τ>τ(q)$, where we give an explicit equation that $τ(q)$ solves.