Cluster volumes for the Gaussian free field on metric graphs

TL;DR

This paper investigates the size of critical clusters in the Gaussian free field on metric graphs, confirming conjectured volume growth rates below the upper-critical dimension and providing tail asymptotics.

Contribution

It establishes the volume scaling of critical clusters below dimension 6 and extends results to graphs with polynomial volume growth and Green's function decay.

Findings

Largest cluster volume scales as r^{(d+2)/2} below dimension 6

Precise tail asymptotics for the cluster volume at the origin

Lower bounds on near-critical cluster volume tails

Abstract

We study the volume of the critical clusters for the percolation of the level sets of the Gaussian free field on metric graphs. On $\mathbb{Z}^d$ below the upper-critical dimension $d=6$, we show that the largest such cluster in a box of side length $r$ has volume of order $r^{\frac{d+2}{2}}$, as conjectured by Werner in arXiv:2002.11487. This is in contrast to the mean-field regime $d>6$, where this volume is of order $r^4$. We further obtain precise asymptotic tails for the volume of the critical cluster of the origin, and a lower bound on the tail of the volume of the near-critical cluster of the origin below the upper-critical dimension. Our proof extends to any graph with polynomial volume growth and polynomial decay of the Green's function as long as the critical one-arm probability decays as the square root of the Green's function, which is satisfied in low enough dimension.