IS04: Geometry of random walks

date: 7/14/2025, time: 14:00-15:30, room: IM EM

Organizer: Bruno Schapira (Aix-Marseille University)

Chair: Bruno Schapira (Aix-Marseille University)

Three-dimensional loop-erased random walks

Sarai Hernandez-Torres (Universidad Nacional Autónoma de México)

The loop-erased random walk (LERW) is a fundamental model for random self-avoiding curves. Since its introduction by Lawler in the 1980s, the scaling limits of LERW have been thoroughly studied. While these limits are well-understood in dimensions two, four, and higher, the three-dimensional case continues to present unique challenges.

This talk will present sharp estimates for the one-point function for the LERW on the integer lattice \(\mathbb{Z}^3\). We will focus on the interplay between the discrete setting and the properties arising in the scaling limit. This talk is based on joint work with Xinyi Li and Daisuke Shiraishi.

Bibliography

\([1]\) S. Hernandez-Torres, X. Li and D. Shiraishi “Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk" Preprint, available at arXiv:2403.07256.

Local geometry of a confined random walk through tilted interlacements

Nicolas Bouchot (Université Paris Dauphine-PSL)

The study of the local geometry of the simple random walk (SRW) on \(\mathbb{Z}^d, d \geq 3\), in high-density regime was somewhat initiated through the works of Benjamini & Sznitman \([1]\) on the SRW on the discrete torus. Sznitman \([3]\) then introduced random interlacements as an “infinite volume limit” of this SRW in the form of a Poisson cloud of SRW trajectories in \(\mathbb{Z}^d\). In this talk, I will present a work about another way to achieve such high-density regime: trapping the random walk inside a large domain of \(\mathbb{Z}^d\). This confined RW can be expressed as a RW on conductances given by the principal (discrete) Laplace eigenfunction. I will show how one can do a local coupling between this confined walk and random interlacements that are locally tilted by this eigenfunction, as well as an application for computing the covering time of an inner subset of the domain.

Bibliography

\([1]\) Itai Benjamini & Alain-Sol Sznitman. "Giant component and vacant set for random walk on a discrete torus." Journal of the European Mathematical Society, vol. 10, no. 1, 2010, pp. 133-172.

\([2]\) Nicolas Bouchot. "A confined random walk locally looks like tilted random interlacements." arXiv:2405.14329.

\([3]\) Alain-Sol Sznitman. "Vacant set of random interlacements and percolation." Annals of Mathematics, vol. 171, no. 3, 2010, pp. 2039-2087.

Random Walk among Moving Traps

Alexander Drewitz (Universität zu Köln)

Random walk and Brownian motion among static obstacles have been the subject of extensive research during the last couple of decades. Of particular interest have been the survival probability as well as the path behavior of the motion conditioned on survival. However, comparatively less attention has been given to scenarios involving moving traps. We survey some of the results obtained in the setting of a Poisson cloud of moving traps during the last decade and discuss a recently established functional central limit theorem in dimensions 6 and higher.