Spanning tree invariants, loop systems and doubly stochastic matrices

The spanning tree invariant of Lind and Tuncel [12] is observed in the context of loop systems of Markov chains. For n=1,2,3 the spanning tree invariants of the loop systems of a Markov chain determined by an irreducible stochastic (n×n)-matrix P coincide if and only if P is doubly stochastic and, in this case, the common value of the spanning tree invariants of the loop systems is n.