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

Abstract

The study of Gaussian free field level sets on supercritical Galton-Watson trees has been initiated by Abächerli and Sznitman in Ann. Inst. Henri Poincaré 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 $h_{\ast }$ 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 $h_{\ast }.$ 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.

arXiv comment38 pages 1 figure

References in corpus (14)

• Phase transition and level-set percolation for the Gaussian free field
• From loop clusters and random interlacement to the free field
• Equality of critical parameters for percolation of Gaussian free field level-sets
• The effect of small quenched noise on connectivity properties of random interlacements
• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• On the transience of processes defined on Galton--Watson trees
• On the radius of Gaussian free field excursion clusters
• Geometry of Gaussian free field sign clusters and random interlacements
• Cluster capacity functionals and isomorphism theorems for Gaussian free fields
• Percolation in the Harmonic Crystal and Voter Model in three dimensions
• Level-set percolation for the Gaussian free field on a transient tree
• Random walks on Galton-Watson trees with random conductances
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Critical and near-critical level-set percolation of the Gaussian free field on regular trees

Cited by in corpus (1)

• Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension