Annealed limit theorems for the ising model on random regular graphs

Can, Van Hao

doi:10.48550/arxiv.1701.08639

Cited by 3 publications

(23 citation statements)

References 18 publications

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“…In [9], we generalize the result in [15] for all random regular graphs, and show that the thermodynamic limits in quenched and annealed models are actually the same. In this paper, we are going to study critical behaviors of the annealed model.…”

Section: Introductionsupporting

confidence: 57%

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Critical behavior of the annealed ising model on random regular graphs

Can

2017

Preprint

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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.

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Section: Introductionsupporting

confidence: 57%

“…Before stating our main results, we first give some definitions following [15,9] of the thermodynamic quantities in finite volume.…”

Section: Introductionmentioning

confidence: 99%

Critical behavior of the annealed ising model on random regular graphs

Can

2017

Preprint

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“…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.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…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].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

et al. 2018

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We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs.where ℓ n = i∈[n] w i is the total weight of all vertices. Denote the law of GRG n (w) by P and its expectation by E. There are many related random graph models (also called rank-1 inhomogeneous random graphs [2]), such as the random graph with specified expected degrees or Chung-Lu model [6,7] and the Poisson random graph or Norros-Reittu model [24]. Janson [21] shows that many of these models are asymptotically equivalent. Even though his results do not apply to the large deviation properties of these random graphs, all our results also apply to these other models.We need to assume that the vertex weight sequences w = (w i ) i∈[n] are sufficiently nicely behaved. Let U n ∈ [n] denote a uniformly chosen vertex in GRG n (w) and W n = w Un its weight. Then, the following condition defines the asymptotic weight W and set the convergence properties of (W n ) n≥1 to W : Condition 1.1 (Weight regularity). There exists a random variable W such that, as n → ∞, (a) W n D −→ W , where D −→ denotes convergence in distribution;

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Annealed Ising model on configuration models

Can¹,

Giardinà²,

Giberti³

et al. 2019

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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

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Annealed limit theorems for the ising model on random regular graphs

Cited by 3 publications

References 18 publications

Critical behavior of the annealed ising model on random regular graphs

Critical behavior of the annealed ising model on random regular graphs

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

Annealed Ising model on configuration models

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