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.
Bounds using hitting time were given in Matthews @cite_17 , @math where @math is the expected time for the walk starting at @math to hit @math .
@cite_17
2138144746 2136645553 2170807097 2949555217
A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given. This technique is used to prove (i) A tight bound of @math for the two-dimensional torus; (ii) A tight bound of @math for trees with maximum degree @math ; (iii) Tight bounds of @math for rapidly mixing walks on vertex transitive graphs, where @math denotes the maximum expected hitting time between vertices.In addition to these new results, the technique allows several known lower bounds on cover times to be systematically proved, often in a much simpler way.Finally, a different technique is used to prove an @math lower bound on the cover time, where @math is the second largest eigenvalue of the transition matrix. This was previously known only in the case where the walk starts in the stationary distribution [... We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds (P. Matthews, 1988) on the cover time, C, and prove that M 2<C= O(M(lnlnn) sup 2 ). We give a deterministic-polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((lnlnn) sup 2 ), improving the previous bound O(lnn) due to Matthews. The blanket time B was introduced by P. Winkler and D. Zuckerman (1996): it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C spl les B. Winkler and Zuckerman conjectured B=O(C) and proved B=O(Clnn). Our bounds above are also valid for the blanket time, and so it follows that B=O(C(lnlnn) sup 2 ). We prove that the expected time for a random walk to cover all n vertices of a graph is at least (1 + o(1))n In n. © 1995 Wiley Periodicals, Inc. 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 1 as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009).