Scaling limits for the critical level-set percolation of the Gaussian free field on regular trees

TL;DR

This paper advances the understanding of critical level-set percolation for the discrete Gaussian free field on infinite regular trees. It derives sharp asymptotics for the critical one-arm event, establishes a Yaglom-type limit for the field values on the large critical cluster, and proves that, under conditioning, the rescaled critical component converges to Aldous' continuum random tree. The analysis relies on representing the GFF as a branching process with an unbounded type space and employing spine techniques to obtain many-to-few formulas, Yaglom limits, and invariance principles. The results connect percolation phenomena on trees to classical branching-process limits and to the geometry of CRTs, with potential implications for related level-set and random-tree models. Overall, the work provides a coherent triptych of asymptotics, limit theorems, and scaling limits at criticality, enriching the theory of GFF level-set percolation on non-Euclidean graphs.

Abstract

We continue the study of the level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, initiated in arXiv:2302.02753. First, we derive a sharp asymptotic estimate for the probability that the connected component of the critical level set containing the root of the tree reaches generation $n$. In particular, we show that the one-arm exponent satisfies $ρ=1$. Next, we establish a Yaglom-type limit theorem for the values of the GFF at generation $n$ within this component. Finally, we show that, after a correct rescaling, this component conditioned on reaching generation $n$ converges, as $n\to\infty$, to Aldous' continuum random tree.

Theorems & Definitions (78)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Proposition 3.1: Szn15 Propositions 3.1, 3.3, Corollary 4.5
• Proposition 3.2
• Proposition 3.3: Theorem 2.1 and Theorem 2.3 in CerLoc23
• Lemma 3.4
• proof
• Proposition 3.5