In this paper we study the time required for a @math -biased ( @math ) walk to visit all nodes of a supercritical Galton-Watson tree up to generation @math . Inspired by the extremal landscape approach in [ 2018] for simple random walk on binary trees, we establish the near-independent nature of extremal points for the @math -biased walk, and deduce the scaling limit of the cover time.
LetT (x;") denote the rst hitting time of the disc of radius " centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x2T2T (x;")=j log"j 2 ! 2= as " ! 0. The same applies to Brownian motion on any smooth, compact connected, two- dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 is asymptotic to 4n 2 (logn) 2 = . Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied non-rigorously in the physics literature. We also establish a conjecture, due to Kesten and R ev esz, that describes the asymptotics for the number of steps needed by simple random walk in Z 2 to cover the disc of radius n. We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length @math . Dembo, Peres, Rosen, and Zeitouni [Ann. Math. 160:433-464, 2004] identified the leading term in the asymptotics for the cover time as @math goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler [Probab. Theory Relat Fields. 167:461-552, 2017]. The ( )-cover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance ( >0 ) from any point. Its leading order in the small ( )-regime has been established by (Ann Math 160:433–464, 2004). In this work, the second order correction is identified. The approach relies on a multi-scale refinement of the second moment method, and draws on ideas from the study of the extremes of branching Brownian motion. Let @math denote the cover time of a two dimensional compact manifold @math by a Wiener sausage of radius @math . We prove that @math is tight, where @math denotes the area of @math . We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus,and show that in both cases with probability approaching @math as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progresstowards a conjecture of Bramson and Zeitouni (2009).