Probability and Mathematical Physics Seminar

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Percolation of discrete GFF in dimension two

Speaker: Pierre Nolin, City University of Hong Kong

Location: Warren Weaver Hall 1302

Date: Friday, May 8, 2026, 12:10 p.m.

Synopsis:

We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in dimension two. For a DGFF $\varphi$ defined in a box with side length $N$, we show that with probability tending to $1$ as $N \to \infty$, there exist "low" crossings, along which $|\varphi| \le C \sqrt{\log \log N}$, for $C$ large enough (while the average and the maximum of $\varphi$ are of order $\sqrt{\log N}$ and $\log N$, respectively). 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 $\alpha=1/2$. We further extend our study to the occupation field of the RWLS for all subcritical intensities $\alpha\in(0,1/2)$, 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 four preprints https://arxiv.org/abs/2409.16230, https://arxiv.org/abs/2409.16273, https://arxiv.org/abs/2504.06202, and https://arxiv.org/abs/2509.25024, all joint with Yifan Gao and Wei Qian (as well as Yijie Bi for the last one).