Recurrence of Markov chain traces
release_giwgwwitc5emlitkwcwymu57by
by
Itai Benjamini, Jonathan Hermon
2017
Abstract
It is shown that transient graphs for the simple random walk do not admit a
nearest neighbor transient Markov chain (not necessarily a reversible one),
that crosses all edges with positive probability, while there is such chain for
the square grid Z^2. In particular, the d-dimensional grid
Z^d admits such a Markov chain only when d=2. For d=2 we present
a relevant example due to Gady Kozma, while the general statement for transient
graphs is obtained by proving that for every transient irreducible Markov chain
on a countable state space, which admits a stationary measure, its trace is
a.s. recurrent for simple random walk. The case that the Markov chain is
reversible is due to Gurel-Gurevich, Lyons and the first named author (2007).
We exploit recent results in potential theory of non-reversible Markov chains
in order to extend their result to the non-reversible setup.
In text/plain
format
Archived Files and Locations
application/pdf 408.0 kB
file_35fg2ayfvnf5bot3xwczlzaf3a
|
arxiv.org (repository) web.archive.org (webarchive) |
1711.03479v1
access all versions, variants, and formats of this works (eg, pre-prints)