During Michaelmas Term, OxTalks will be moving to a new platform (full details are available on the Staff Gateway).
For now, continue using the current page and event submission process (freeze period dates to be advised).
If you have any questions, please contact halo@digital.ox.ac.uk
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.