Singularities of the Effective Fugacity on the Ising Model on the Cayley Tree

Abstract

. Evidence suggesting a connection between Lee--Yang singularities of the magnetization of the Ising model and the singularities of the effective fugacity is presented. 1 Introduction The study of the Griffiths' singularities in random ferromagnetic Ising models [2] (see [3] and [4] for general discussions) remains one of the most living topics of research in Statistical Mechanics of spin systems. Besides its physical relevance, it involves various mathematical difficulties that make a detailed analysis particularly challenging. One of the main difficulties relies in the almost absence of models where a concrete study of the analytic properties of the magnetization and other thermodynamic quantities, as functions of the (complex) fugacity, could be developed. In [6] a detailed rigorous analysis of the presence of Griffiths' singularities has been performed in a class of layered diluted ferromagnetic Ising models on the Cayley tree. This involved the study of the analytic structure of t..