This talk concerns the simple random walk on a two-dimensional torus of large size, and its continuous counterpart, Brownian motion. We discuss large deviation results for various covering events on the torus, in the line of the classical work of Dembo, Peres, Rosen and Zeitouni on cover times. We also present the random interlacements model on ℤ², introduced by Sznitman in dimensions d ≥ 3 and adapted to the two-dimensional setting by Comets, Popov and Vachkovskaia, which arises as a local limit in the neighbourhood of a site left unvisited by the walk. We describe some properties of this dependent percolation model in dimension two. Historically, such questions have been studied in all dimensions, but dimension two is a special case, due to the recurrence of the simple random walk.