Let X be a simple random walk in \mathbb{Z}_n^d with d\geq 3 and let t_{\rm{cov}} be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set \mathcal{L}_\alpha of points that have not been visited by time \alpha t_{\rm{cov}} and prove that it exhibits a phase transition: there exists \alpha_* so that for all \alpha>\alpha_* and all \epsilon>0 there exists a coupling between \mathcal{L}_\alpha and two i.i.d. Bernoulli sets \mathcal{B}^{\pm} on the torus with parameters n^~~d}with the property that \mathcal{B}^~~\subseteq \mathcal{L}_\alpha\subseteq \mathcal{B}^+ with probability tending to 1 as n\to\infty. When \alpha\leq \alpha_*, we prove that there is no such coupling.

**Date**: 15 January 2024, 14:00 (Monday, 1st week, Hilary 2024)**Venue**: Mathematical Institute

Woodstock Road OX2 6GGSee location on maps.ox**Details**: L5**Speaker**: Perla Sousi (University of Cambridge)**Organising department**: Department of Statistics**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Public- Editors: James Martin, Julien Berestycki