Monotonicity of Ursell functions in the Ising model

TL;DR

This paper proves a monotonicity property of Ursell functions in ferromagnetic Ising models and applies it to confirm a 1983 conjecture about the movement of zeros of the partition function with increasing interactions.

Contribution

It establishes a new monotonicity property of Ursell functions in the Ising model and uses it to prove a longstanding conjecture about the zeros of the partition function.

Findings

Ursell functions $u_{2k}$ satisfy $(-1)^{k-1}u_{2k}$ increasing in each interaction.

The closest zero of the partition function moves towards the origin as interactions increase.

Confirmed a 1983 conjecture by Nishimori and Griffiths.

Abstract

In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions $u_{2k}$ satisfy: $(-1)^{k-1}u_{2k}$ is increasing in each interaction. As an application, we prove a 1983 conjecture by Nishimori and Griffiths about the partition function of the Ising model with complex external field $h$: its closest zero to the origin (in the variable $h$) moves towards the origin as an arbitrary interaction increases.