Critical and near-critical level-set percolation of the Gaussian free field on regular trees

TL;DR

This paper investigates the percolation properties of the Gaussian free field's level sets on regular trees, focusing on critical and near-critical regimes, providing new insights into their connectivity and tail behaviors.

Contribution

It establishes the continuity of percolation probability, derives tail estimates for component sizes, and characterizes near-critical asymptotics for the Gaussian free field on regular trees.

Findings

Percolation probability is continuous at criticality.

Exact tail estimates for the size of critical level set components.

Asymptotic behavior of percolation probability in near-critical regime.

Abstract

For the Gaussian free field on a $(d + 1)$-regular tree with $d \geq 2$, we study the percolative properties of its level sets in the critical and the near-critical regime. In particular, we show the continuity of the percolation probability, derive an exact symptotic tail estimate for the cardinality of the connected component of the critical level set, and describe the asymptotic behaviour of the percolation probability in the near-critical regime.