Limiting behavior for the distance of a random walk
Nathanael Berestycki, University of Cambridge
Rick Durrett, Cornell University
Abstract
In this paper we study some aspects of the behavior of random
walks on large but finite graphs before they have reached their
equilibrium distribution. This investigation is motivated by a
result we proved recently for the random transposition random
walk: the distance from the starting point of the walk has a phase
transition from a linear regime to a sublinear regime at time
$n/2$. Here, we study the examples of random 3-regular graphs,
random adjacent transpositions, and riffle shuffles. In the case
of a random 3-regular graph, there is a phase transition where the
speed changes from 1/3 to 0 at time $3log_2 n$. A similar result
is proved for riffle shuffles, where the speed changes from 1 to 0
at time $log_2 n$. Both these changes occur when a distance equal
to the average diameter of the graph is reached. However in the
case of random adjacent transpositions, the behavior is more
complex. We find that there is no phase transition, even though
the distance has different scalings in three different regimes.
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