Abstract
We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by N) versions \(D_N\subseteq {\mathbb {Z}}^2\) of bounded open domains \(D\subseteq {\mathbb {R}}^2\). Upon exit from \(D_N\), the walk lands on a “boundary vertex” and then reenters \(D_N\) through a random boundary edge in the next step. In the parametrization by the local time at the “boundary vertex” we prove that, at times corresponding to a \(\theta \)-multiple of the cover time of \(D_N\), the sets of suitably defined \(\lambda \)-thick (i.e., heavily visited) and \(\lambda \)-thin (i.e., lightly visited) points are, as \(N\rightarrow \infty \), distributed according to the Liouville Quantum Gravity \(Z^D_\lambda \) with parameter \(\lambda \)-times the critical value. For \(\theta <1\), also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to \(Z^D_{\sqrt{\theta }}\). The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.
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Acknowledgements
The first author has been supported in part by JSPS KAKENHI, Grant-in-Aid for Early-Career Scientists 18K13429. The second author has been partially supported by the NSF award DMS-1712632. We wish to thank an anonymous referee for a number of important corrections to the initial version of this manuscript.
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Abe, Y., Biskup, M. Exceptional points of two-dimensional random walks at multiples of the cover time. Probab. Theory Relat. Fields 183, 1–55 (2022). https://doi.org/10.1007/s00440-022-01113-4
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DOI: https://doi.org/10.1007/s00440-022-01113-4