Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings Via Transfer-Impedances

Abstract

Let $G$ be a finite graph or an infinite graph on which $\mathbb{Z}^d$ acts with finite fundamental domain. If $G$ is finite, let $\mathbf{T}$ be a random spanning tree chosen uniformly from all spanning trees of $G$; if $G$ is infinite, methods from Pemantle show that this still makes sense, producing a random essential spanning forest of $G$. A method for calculating local characteristics (i.e., finite-dimensional marginals) of $\mathbf{T}$ from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix ($n$-dimensional minors) are needed to compute small ($n$-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for $G$, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when $G$ is an infinite graph admitting a $Z^d$-action with finite quotient. The same computation also gives the entropy of the law of $\mathbf{T}$. These results are applied to the problem of tiling certain lattices by dominos--the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to tree-valued random variables and criteria for these generalized moments to determine a distribution.