Generating Galton-Watson trees using random walks and percolation for the Gaussian free field

TL;DR

This paper extends the understanding of Gaussian free field level sets on supercritical Galton-Watson trees, proving positivity of the percolation critical parameter and demonstrating how positive correlations aid percolation, using a novel exploration via random walks.

Contribution

It generalizes previous results to arbitrary offspring distributions and random conductances, providing a rigorous proof of the positivity of the critical parameter and insights into correlation effects on percolation.

Findings

Positivity of the percolation critical parameter $h_*$ established.

Random interlacement set on Galton-Watson trees shown to be transient.

Level sets of Gaussian free field above small levels are shown to percolate.

Abstract

The study of Gaussian free field level sets on supercritical Galton-Watson trees has been initiated by Ab\"acherli and Sznitman in Ann. Inst. Henri Poincar\'{e} Probab. Stat., 54(1):173--201, 2018. By means of entirely different tools, we continue this investigation and generalize their main result on the positivity of the associated percolation critical parameter $h_*$ to the setting of arbitrary supercritical offspring distribution and random conductances. A fortiori, this provides a positive answer to the open question raised at the end of the aforementioned article. What is more, in our setting it also establishes a rigorous proof of the physics literature mantra that positive correlations facilitate percolation when compared to the independent case. Our proof proceeds by constructing the Galton-Watson tree through an exploration via finite random walk trajectories. This exploration of the tree progressively unveils an infinite connected component in the random interlacements set on the tree, which is stable under small quenched noise. Using a Dynkin-type isomorphism theorem, we then infer the strict positivity of the critical parameter $ h_* .$ As a byproduct of our proof we obtain the transience of the random interlacement set and the level sets of the Gaussian free field above small positive levels on such Galton-Watson trees.