356 research outputs found
Random walks on trees
AbstractThe classical gambler's ruin problem, i.e., a random walk along a line may be viewed graph theoretically as a random walk along a path with the endpoints as absorbing states. This paper is an investigation of the natural generalization of this problem to that of a particle walking randomly on a tree with the endpoints as absorbing barriers. Expressions in terms of the graph structure are obtained from the probability of absorption at an endpoint e in a walk originating from a vertex v, as well as for the expected length of the walk
The Best Mixing Time for Random Walks on Trees
We characterize the extremal structures for mixing walks on trees that start
from the most advantageous vertex. Let be a tree with stationary
distribution . For a vertex , let denote the expected
length of an optimal stopping rule from to . The \emph{best mixing
time} for is . We show that among all trees with
, the best mixing time is minimized uniquely by the star. For even ,
the best mixing time is maximized by the uniquely path. Surprising, for odd
, the best mixing time is maximized uniquely by a path of length with
a single leaf adjacent to one central vertex.Comment: 25 pages, 7 figures, 3 table
The most visited sites of biased random walks on trees
We consider the slow movement of randomly biased random walk on a
supercritical Galton--Watson tree, and are interested in the sites on the tree
that are most visited by the biased random walk. Our main result implies
tightness of the distributions of the most visited sites under the annealed
measure. This is in contrast with the one-dimensional case, and provides, to
the best of our knowledge, the first non-trivial example of null recurrent
random walk whose most visited sites are not transient, a question originally
raised by Erd\H{o}s and R\'ev\'esz [11] for simple symmetric random walk on the
line.Comment: 17 page
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