Abstract:
We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in dimension two. For a DGFF 
defined in a box with side length 
, we show that with probability tending to 
polynomially fast in , there exist “low’’ crossings, along which 
, for any 
(while the average and the maximum of are of order 
and 
, respectively). Our method also strongly suggests the existence of such crossings below 
, for 
large enough. As a consequence, we obtain connectivity properties for 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 
. We further extend our study to the occupation field of the RWLS for all subcritical intensities 
, and in that case we uncover a non-trivial phase transition. This work relies heavily on new tools and techniques that we developed for the RWLS, especially surgery arguments on loops, which were made possible by a separation result for random walks in a loop soup. This allowed us to obtain a precise upper bound for the probability that two large connected components of loops “almost touch”, which is instrumental here.
This talk is based on the two preprints https://arxiv.org/abs/2409.16230 and https://arxiv.org/abs/2409.16273.
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