Spanning trees in two dimensions

Abstract

We describe the results of an analytical and numerical study of the geometrical properties of random spanning trees on a square lattice. We determine exactly the probability distribution of the coordination number at a site on a random spanning tree. We argue that the probability that s sites get disconnected from the tree on deleting a bond at random from the tree varies as s for large s. The probability that a loop of perimeter l is formed on adding an additional link at random varies as l for large l. These distributions are also determined numerically in a Monte Carlo simulation on random spanning trees generated by using Broder's algorithm. The numerical results are in complete agreement with the theoretical predictions.