I shall outline the proof of Aldous's (1989) conjecture that it takes a
simple random walk about $(N/\pi) \log N$ steps to hit all $N \gg 1$ points of a
(large) planar lattice torus. Along the way, I shall provide the asymptotics of the
amount of time needed for the Wiener sausage of radius $r \to 0$ to completely cover a
smooth, compact, connected, two-dimensional, Riemannian manifold without boundary.
Fixing $0
A common theme is the `tree like' correlation structure of the occupation measure, used for a
`multi-scale refinement' of the second moment method. This talk is based on joint works with
Yuval Peres, Jay Rosen and Ofer Zeitouni.