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Phase transition for the late points of random walk
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
Venue:
Mathematical Institute, Woodstock Road OX2 6GG
Venue 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
