  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.
