Random spanning forests, Markov matrix spectra and well distributed points

TL;DR

This paper explores the use of random spanning forests to achieve uniform hitting times in Markov processes, investigates their spectral properties, and extends related theorems to non-reversible cases, connecting probabilistic and algebraic methods.

Contribution

It introduces a novel probability law on subsets for Markov processes with uniform hitting times and links spanning forests to generator spectra, extending key theorems to non-reversible chains.

Findings

Established a probability law on subsets with uniform mean hitting times.

Connected spanning forests to the spectrum of Markov generators.

Extended Burton and Pemantle transfer current theorem to non-reversible Markov chains.

Abstract

This paper is a variation on the uniform spanning tree theme. We use random spanning forests to solve the following problem: for a Markov process on a finite set of size $n$, find a probability law on the subsets of any given size $m \leq n$ with the property that the mean hitting time of such a random target does not depend on the starting point of the random walk. We then explore the connection between random spanning forests and infinitesimal generator spectrum. In particular we give an almost probabilistic proof of an algebraic result due to Micchelli and Willoughby and used by Fill and Miclo to study the convergence to equilibrium of reversible Markov chains. We finally introduce some related fragmentation and coalescence processes, emphasizing algorithmic aspects, and give an extension of Burton and Pemantle transfer current theorem to the non reversible case.