IS04: Geometry of Random Walks

Organizer: Bruno Schapira (Aix-Marseille University)

Three-dimensional loop-erased random walks

Sarai Hernandez-Torres

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.