Recurrence of Markov chain traces [article]

Itai Benjamini, Jonathan Hermon
2019 arXiv   pre-print
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
more » ... le 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.
arXiv:1711.03479v4 fatcat:ong6of5xajbmxb7bokxmaawr2m