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

Abstract

We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\phi$ defined in a box $B_N$ with side length $N$, for $C$ large enough, there exist low crossings in the set of vertices $z$ where $|\phi (z)|\le C\sqrt{\log \log N}$, with probability tending to $1$ as $N\to \infty$, while the average and the maximum of $\phi$ are of order $\sqrt{\log N}$ and $\log N$, respectively. 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.

71 pages, 9 figures; improved main results, and additional simulation for the threshold distribution