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 ${\mathbb{Z}}^d$, $d\ge 3$, are analytic on the whole off-critical regime $\mathbb{R}\setminus \{h_{\ast }\}$. 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 $h\ne h_{\ast }$, which we believe to be a result of independent interest. We also discuss the case of general transient graphs.

arXiv comment32 pages

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Cited by in corpus (4)

• Equality of critical parameters for percolation of Gaussian free field level-sets
• On the radius of Gaussian free field excursion clusters
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• First passage percolation, local uniqueness for interlacements and capacity of random walk