Percolation of discrete GFF in dimension two II. Connectivity properties of two-sided level sets

TL;DR

This paper investigates the connectivity of two-sided level sets in the 2D discrete Gaussian free field and related loop soup models, revealing phase transitions and percolation properties using advanced probabilistic techniques.

Contribution

It establishes new connectivity results for level sets of the DGFF and RWLS, demonstrating phase transitions and extending analysis to subcritical intensities.

Findings

Existence of low crossings in DGFF level sets with high probability

Identification of a phase transition in the RWLS occupation field

Connectivity of the 'carpet' set in the loop soup model

Abstract

We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\varphi$ defined in a box $B_N$ with side length $N$, we show that with high probability, there exist low crossings in the set of vertices $z$ with $|\varphi(z)|\le\varepsilon\sqrt{\log N}$ for any $\varepsilon>0$, while the average and the maximum of $\varphi$ are of order $\sqrt{\log N}$ and $\log N$, respectively. Our method also strongly suggests the existence of such crossings below $C\sqrt{\log\log N}$, for $C$ large enough. As a consequence, we also obtain connectivity properties of the set of thick points of a random walk. We rely on an isomorphism between the DGFF and the random walk loop soup (RWLS) with critical intensity $\alpha=1/2$, and further extend our study to the occupation field of the RWLS for all subcritical intensities $\alpha\in(0,1/2)$. For the RWLS in $B_N$, we show that for $\lambda$ large enough, there exist low crossings of $B_N$, remaining below $\lambda$, even though the average occupation time is of order $\log N$. Our results thus uncover a non-trivial phase-transition for this highly-dependent percolation model. For both the DGFF and the occupation field of the RWLS, we further show that such low crossings can be found in the "carpet" of the RWLS - the set of vertices which are not in the interior of any cluster of loops. This work is the second part of a series of two papers. It relies heavily on tools and techniques developed for the RWLS in the first part, especially surgery arguments on loops, which were made possible by a separation result in the RWLS. This allowed us, in that companion paper, to derive several useful properties such as quasi-multiplicativity, and obtain a precise upper bound for the probability that two large connected components of loops "almost touch", which is instrumental here.