Cover time and favourite points for planar random walks

What is the number of steps it takes a Simple Random Walk (SRW) to cover all $N >> 1$ points of a (large) planar lattice torus? Alternatively, what is the asymptotics of the number of steps needed for planar SRW to cover the disc of radius $n$? A dual question is what is the number of visits by a planar SRW to the lattice site most frequently visited during its first $n$ steps? In the talk we answer these questions, resolving in the process long standing conjectures of Aldous-Lawler, Kesten-Revesz and Erdos-Taylor. Key to our proofs is the formulation and study of analogous problems involving the occupation measure of small discs by a planar Brownian motion. The latter exhibit ``tree like'' correlation structure, giving rise to a ``multiscale refinement'' of the second moment method. This talk is based on a joint work with Yuval Peres, Jay Rosen and Ofer Zeitouni.