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 $${{\mathbb {Z}}}^d$$ Z d , $$d\ge 3$$ d ≥ 3 , are analytic on the whole off-critical regime $${{\mathbb {R}}}\setminus \{h_*\}$$ R \ { h ∗ } . This result concerns in particular the percolation density function $$\theta (h)$$ θ ( h ) and the (truncated) susceptibility $$\chi (h)$$ 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 $$h\ne h_*$$ h ≠ h ∗ , which we believe to be a result of independent interest. We also discuss the case of general transient graphs.