On cover times for 2D lattices

We study the cover time ${\tau }_{\mathrm{c}\mathrm{o}\mathrm{v}}$ 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 1 as $n$ increases, $\sqrt{{\tau }_{\mathrm{c}\mathrm{o}\mathrm{v}}}=\sqrt{2n^2}[\sqrt{2\mathrm{/}\pi }\log n+O(\log \log n)]$. This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009).