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

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 $d\ge 3$ 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 $q\ge 2$ 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.

27 pages. Comments welcome