Annealed central limit theorems for the ising model on random graphs

C. Giardinà, C. Giberti, R.W. van der Hofstad, M.L. Prioriello

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Abstract

The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by √N of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature 0≤ βan n <∞ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature and external magnetic field B for which either β <βan n and B = 0, or β > 0 and B ≠ 0. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters β and B, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.

Original languageEnglish
Pages (from-to)121-161
Number of pages41
JournalAlea : Latin American journal of probability and mathematical statistics
Volume13
Issue number1
Publication statusPublished - 2016

Keywords

  • Annealed measure
  • Central limit theorem
  • Ising model
  • Random graphs

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