# Weak Mixing and Analyticity of the Pressure in the Ising Model

## TL;DR

This paper establishes the analyticity of the pressure in the ferromagnetic Ising model under weak mixing conditions and proves weak mixing when the magnetic field is non-zero, using graphical and cluster expansion methods.

## Contribution

It demonstrates the analyticity of the pressure in the Ising model under weak mixing and proves weak mixing for non-zero magnetic fields, extending known results.

## Key findings

- Pressure is analytic when the model has exponential weak mixing.
- Weak mixing holds whenever the magnetic field is non-zero.
- Analyticity and weak mixing are valid outside the critical transition line.

## Abstract

We prove that the pressure (or free energy) of the finite range ferromagnetic Ising model on $\mathbb{Z}^d$ is analytic as a function of both the inverse temperature $\beta$ and the magnetic field $h$ whenever the model has the exponential weak mixing property. We also prove the exponential weak mixing property whenever $h\neq 0$. Together with known results on the regime $h=0,\beta<\beta_c$, this implies both analyticity and weak mixing in all the domain of $(\beta,h)$ outside of the transition line $[\beta_c,\infty)\times \{0\}$. The proof of analyticity uses a graphical representation of the Glauber dynamic due to Schonmann and cluster expansion. The proof of weak mixing uses the random cluster representation.