Uniform analyticity of local observables in FK-percolation and analyticity of the Ising spontaneous magnetisation

TL;DR

This paper establishes uniform analyticity of local observables in FK-percolation and derives the analyticity of the Ising model's spontaneous magnetisation across various regimes and dimensions.

Contribution

It proves uniform analyticity of local event probabilities in FK-percolation and applies this to show magnetisation and susceptibility are analytic in the Potts and Ising models.

Findings

Probabilities of local events are uniformly analytic in the percolation parameter p.

Magnetisation of the Potts model is analytic in a broad parameter range, including the entire supercritical regime for the Ising case.

Susceptibility of the Potts model is analytic in the subcritical interval.

Abstract

We prove that, in the FK-percolation model, the probabilities of local events are uniformly analytic in the percolation parameter $p$ under suitable mixing assumptions on the measure, and satisfy a uniform exponential growth bound. This result allows us to prove that the magnetisation of the Potts model is analytic in a suitable range of parameters, including the Ising case in all dimensions $d \geq 3$ in the whole supercritical regime. We also provide a proof of the analyticity of the susceptibility of the Potts model with $q$ colours, for any $q \geq 2$ in the whole subcritical interval. Finally, we prove the analyticity of various quantities in the FK-percolation measure, including the multi-point and truncated multi-point connectivity probabilities.