Analyticity of Gaussian free field percolation observables

Abstract

We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice Z d , d ≥ 3, are analytic on the whole off-critical regime R \ {h * }. This result concerns in particular the percolation density function θ(h) and the (truncated) susceptibility χ(h).As an important step towards the proof, we show the exponential decay in probability for the capacity of a finite cluster for all h = h * , which we believe to be a result of independent interest. We also discuss the case of general transient graphs.