Critical behavior of the annealed ising model on random regular graphs

Abstract

In [15], the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in [9], we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by n 3/4 converges to a specific random variable, with n the number of vertices of random regular graphs.

Large Deviations for the Annealed Ising Model on Inhomogeneous Random Graphs: Spins and Degrees

“…Depending on the setting, the annealed setting may have a different critical temperature. However, as predicted by the non-rigorous physics work [23,14], the annealed Ising model turns out to be in the same universality class as the quenched model for all settings investigated [5,11,13].…”

“…While much work exists on random graphs with independent randomness on the edges or vertices, such as percolation and first-passage percolation (see [20] for a substantial overview of results for these models on random graphs), the dependence of the random variables on the vertices raises many interesting new questions. We refer to [4,5,8,11,12,13,18,17] for recent results on the Ising model on random graphs, as well as [20,Chapter 5] and [9] for overviews. The crux about the Ising model is that the variables that are assigned to the vertices of the random graph wish to be aligned, thus creating positive dependence.…”

“…Since the Ising model lives on a random graph, we are dealing with non-trivial double randomness of both the spin system as well as the random environment. While [8,12,13,17] study the quenched setting, in which the random graph is either fixed (random-quenched) or the Boltzmann-Gibbs measure is averaged out with respect to the random medium (averaged-quenched), recently the annealed setting, in which both the partition function and the Boltzmann weight are averaged out separately has attracted substantial attention [4,5,11,18]. The random graph models investigated are rank-1 inhomogeneous random graphs [11,18], as well as random regular graphs and configuration models [4,5,17].…”

Annealed Ising model on configuration models

“…The case where D = r corresponds to random regular graphs, and is the only case where Theorems 1.2 and 1.5 actually agree, which explains why this case needs to be excluded in Theorem 1.6. The random regular graph case was also investigated by Dommers et al in [16] for r = 2, by Can in [3,4], and by Dembo, Montanari, Sly and Sun [9] (see also the remark below [9, Theorem 1], where it is mentioned that the Ising result also holds for odd degree).…”

“…The annealed Ising model could be solved on two specific random graphs models: the generalized random graph [10], where edges are independent, and the random regular graph [3,4], where the degrees are all equal (see also [16] where the configuration model with degrees that are either 1 or 2 was studied by mapping this model to a one-dimensional Ising model). In this paper, we extend the analysis of the annealed Ising model to the configuration model, using ideas from, and extending work of, Can [3,4]. The model includes degree variability, and dependence between edges, since the edges need to realize a prescribed degree sequence.…”