Resonances in the oriented uniform spanning forest on a two-dimensional torus

2007. január 3. szerda 17:15

Consider an n by m discrete torus with a directed graph structure, in which two edges, pointing north or east, emanate from each vertex. We discuss the uniform spanning forest of this graph, whose characteristics depend sensitively on the aspect ratio m/n of the torus. The expected total number of edges contained in cycles, for example, is peaked when m/n is close to a small rational. This work, joint with Rick Kenyon, complements an earlier paper of Kenyon and Wilson, that analyses resonance among paths in a model that is equivalent to a honeycomb dimer model on a discrete torus.