Analyticity of Gaussian free field percolation observables

TL;DR

This paper proves the analyticity of percolation observables for the Gaussian free field on high-dimensional lattices outside the critical point, including decay properties of finite clusters and implications for general graphs.

Contribution

It establishes the analyticity of key percolation observables for the Gaussian free field on $ olinebreak bZ^d$, $d extgreater 3$, and introduces exponential decay results for finite cluster capacities.

Findings

Percolation density function $ heta(h)$ is analytic off the critical point.

Susceptibility $oldsymbol{ extit{ ext{χ}}}(h)$ is analytic off the critical point.

Exponential decay in probability for the capacity of finite clusters for all $h eq h_*$.

Abstract

We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice $\mathbb{Z}^d$, $d\geq3$, are analytic on the whole off-critical regime $\mathbb{R}\setminus\{h_*\}$. This result concerns in particular the percolation density function $\theta(h)$ and the (truncated) susceptibility $\chi(h)$. As an important step towards the proof, we show the exponential decay in probability for the capacity of a finite cluster for all $h\neq h_*$, which we believe to be a result of independent interest. We also discuss the case of general transient graphs.