Fluctuations of the Magnetization in the p-Spin Curie–Weiss Model

Abstract

In this paper we study the fluctuations of the magnetization in the p-spin Curie–Weiss model, for p⩾ 3. We provide a complete description of the asymptotic distribution of the magnetization in the p-spin Curie–Weiss model, complementing the well-known results in the 2-spin case. Our results unearth various new phase transitions, such as the existence of a certain ‘critical’ curve in the parameter space, where the limiting distribution of the magnetization is a discrete mixture, with local Gaussian fluctuations around each of the atoms. The number of atoms (mixture components) is either two or three depending on the sign of one of the parameters and the parity of p. Another interesting revelation is the existence of certain ‘special’ points in the parameter space where the magnetization converges to a non-Gaussian limiting distribution at rate N14.