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Register a new userAnnealed limit theorems for the ising model on random regular graphs
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Van Hao Can
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In a recent paper [15], Giardin{`a}, Giberti, Hofstad, Prioriello have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs including the random 2-regular graph. We present a new proof of their results, which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.
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In [17], 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 [11], 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 that the magnetization scaled by n 3/4 converges to a specific random variable, with n the number of vertices of random regular graphs.
We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the cavity prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.
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. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct.
Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $dge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a $d$-regular graph chosen uniformly at random from the collection of all $d$-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random $d$-regular graph, both in the quenched as well as the annealed settings. Let $beta$ be the inverse temperature, $beta_c$ be the critical temperature and $B$ be the external magnetic field. Concerning the annealed measure, we show that for $beta > beta_c$ there exists $hat{B}_c(beta)in (0,infty)$ such that the model is metastable (i.e., the mixing time is exponential in the graph size $n$) when $beta> beta_c$ and $0 leq B < hat{B}_c(beta)$, whereas it exhibits the cut-off phenomenon at $c_star n log n$ with a window of order $n$ when $beta < beta_c$ or $beta > beta_c$ and $B>hat{B}_c(beta)$. Interestingly, $hat{B}_c(beta)$ coincides with the critical external field of the Ising model on the $d$-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists $B_c(beta)$ with $B_c(beta) leq hat{B}_c(beta)$ such that for $beta> beta_c$, the mixing time is at least exponential along some subsequence $(n_k)_{kgeq 1}$ when $0 leq B < B_c(beta)$, whereas it is less than or equal to $Cnlog n$ when $B>hat{B}_c(beta)$. The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.
Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given `far away observations. Several theoretical results (and simple algorithms) are available when their joint probability distribution is Markov with respect to a tree. In this paper we consider the case of sequences of random graphs that converge locally to trees. In particular, we develop a sufficient condition for the tree and graph reconstruction problem to coincide. We apply such condition to colorings of random graphs. Further, we characterize the behavior of Ising models on such graphs, both with attractive and random interactions (respectively, `ferromagnetic and `spin glass).
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