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.
More precise results can be obtained if we restrict to particular graphs. Postponding our topic of trees to the next paragraph, the most studied situation is the two-dimensional torus. The first order estimation of its cover time was determined in @cite_7 , then the result was ameliorated in Ding @cite_12 , Belius and Kistler @cite_9 , Abe @cite_8 , and most recently @cite_1 to the extend that @math where @math is a 2-dimensional manifold with some regularity conditions, @math is the area of @math , and @math is the time for the walk to intersect every ball of radius @math on @math .
@cite_7 @cite_8 @cite_9 @cite_1 @cite_12
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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).