1D Ising models, compound geometric distributions and selfdecomposability

Abstract

It is shown that the inverse of the partition function in 1D Ising model, as a function of the external field, is a product of Fourier transforms of compound geometric distributions. These are random sums (randomly stopped random walks) with the probability of a success depending only on the interaction constant K between sites. Moreover, it is proved that those distributions belong to the Lévy class L of selfdecomposable probability measures, therefore they have the background driving Lévy processes. It is important that the general structure of class L characteristic functions is well known and that it is much more specific than the Lévy-Khintchine formula for infinite divisible variables.