Recurrence of Markov chain traces

TL;DR

This paper investigates the recurrence properties of Markov chain traces on transient graphs, showing that certain Markov chains cannot cross all edges with positive probability unless on specific grids, extending previous reversible cases.

Contribution

It extends the understanding of Markov chain traces by proving non-existence results for certain transient graphs and Markov chains, including non-reversible cases, using potential theory.

Findings

Transient graphs for simple random walk do not admit certain transient Markov chains crossing all edges.

The $d$-dimensional grid $ extbf{Z}^d$ admits such chains only when $d=2$.

Trace recurrence is established for all transient irreducible Markov chains with stationary measure.

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 $\mathbb{Z}^2$. In particular, the $d$-dimensional grid $\mathbb{Z}^d$ 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.