A combinatorial proof of a formula of Biane and Chapuy

Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of $\mathcal{G}$ consist of pairs of trees $(t_i, t_j)$ such that $t_j$ can be obtained from $t_i$ by adding the edge from the root of $t_i$ to the root of $t_j$ and deleting the outgoing edge from the root of $t_j$. A formula for the ratio of the sum of the weights of the directed rooted spanning trees on $\mathcal{G}$ to the sum of the weights of the directed rooted spanning trees on $G$ was recently given by Biane and Chapuy. We provide an alternative proof of this formula, which is both simple and combinatorial. The proof involves working with the stochastic zeta function of an irreducible Markov chain. By generalizing the stochastic zeta function we also recover the general result of Biane and Chapuy which gives a formula for the determinant of the Schr\"odinger matrix on $\mathcal{G}$ corresponding to a given Schr\"odinger matrix on $G$, in terms of the minors of the latter matrix.