Fluctuations of the magnetization for Ising models on Erdős–Rényi random graphs—the regimes of small p and the critical temperature *

Abstract

Abstract We continue our analysis of Ising models on the (directed) Erdős–Rényi random graph. This graph is constructed on N vertices and every edge has probability p to be present. These models were introduced and first studied by Bovier and Gayrard (1993 J. Stat. Phys. 72 643–64) and further analyzed by the authors in a previous note, in which we consider the case of p = p(N) satisfying p 3 N 2 → ∞ and β < 1. In the current note we prove a quenched central limit theorem for the magnetization for p satisfying pN → ∞ in the high-temperature regime β < 1. We also show a non-standard central limit theorem for p 4 N 3 → ∞ at the critical temperature β = 1. For p 4 N 3 → 0 we obtain a Gaussian limiting distribution for the magnetization. Finally, in the critical regime p 4 N 3 → c the limiting distribution for the magnetization contains a Gaussian component as well as a e − x 4 -term. Hence, at β = 1 we observe a phase transition in p for the fluctuations of the magnetization.