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Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x∈T 2 T (x, ε)/| log ε| 2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, twodimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to , that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 n is asymptoticdoi:10.4007/annals.2004.160.433 fatcat:jh5ds6z2lbbchlbzlbluh6uexy