On cover times for 2D lattices

Abstract

We study the cover time Tcov by (continuous-time) random walk on the 2D box of side length n with wired boundary or on the 2D torus, and show that in both cases with probability approaching as n increases, This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009).