Random spanning forests, Markov matrix spectra and well distributed points

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.

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