Annealed Ising model on configuration models

Abstract

In this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we take the average on both sides of the ratio defining the Boltzmann-Gibbs measure of the Ising model. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. These results complement several results by the authors, including the setting of the random regular graph by the first author and the setting of the generalized random graph by Dommers, Prioriello and the last three authors. We derive these results by a careful analysis of the annealed partition function in the different cases, using the explicit form of it derived by the first author. In the case of i.i.d. degrees, this is complemented by a large deviation analysis for the empirical degree distribution under the annealed Ising model. c = βD c (recall Remark 1.7). When E[e β an,D c D/2 ] < ∞, then q(β an,D c ) in (1.24) has exponential tails, since cosh(β an,D c ) < e β an,D c . Therefore, denoting q(β, B) to be the empirical degree distribution of the annealed Ising model on the configuration model with i.i.d. degrees, one can expect that for β > β an,D c and B > 0 with β − β an,D c

Glauber dynamics for Ising models on random regular graphs: cut-off and metastability

“…Initially, the focus was on establishing the thermodynamic limit of the quenched Ising model on random graphs [14,13,18] as well as on its critical behavior [19,25]. In the past years, also the annealed Ising model has attracted considerable attention [11,10,12,17,26]. As we explain in more detail below, the quenched and annealed settings for the Ising model describe different physical realities in the dynamics of the underlying graph and the Ising model on it.…”

“…where E is the expectation the random graph under consideration. These two measures concern different physical realities (see also [12]), in the sense that the random graph in the quenched measure is fixed (or can be thought of as varying very slowly compared to the Ising Glauber dynamics), while in the annealed law, the Glauber dynamics only observes an average graph instance, which, by the ergodic theorem, can be thought of as an expectation with respect to the graph randomness. As discussed before, we will work on the d-regular configuration model, but in our discussion we will also discuss more general degree settings.…”