Ramon Locher | Searcharxiv

arXiv2025

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

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 $\rho =1$. 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.

math.PR↗

arXiv2023

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

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.

math.PR↗