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

Abstract

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;

Berry–Esseen bounds in the inhomogeneous Curie–Weiss model with external field

“…Hence, one obtains a two-dimensional regression equation. Looking at the joint distribution of the sum of spins and the weighted sum of spins is for example also used to study their large deviations [5]. Another complication that arises is that the weighted spin sum is not necessarily uniformly bounded.…”

“…In [14], it is proved that in the ICW, and hence also the annealed Ising model on inhomogeneous random graphs, the sum of spins, in the presence of an external field or above the critical temperature, satisfies a central limit theorem. The study of this model continued in [6], where critical exponents were computed and a non-standard limit theorem was obtained at the critical point, and in [5], where large deviations of the sum of spins were studied.…”