SPANNING TREES OF GRAPHS ON SURFACES AND THE INTENSITY OF LOOP-ERASED RANDOM WALK ON PLANAR GRAPHS.

In this article the authors show how to compute the probabilities of various connection topologies for uniform random spanning trees (UST) of graphs on surfaces and the intensity of the loop-erased random walk (LERW) on planar graphs. Topics discussed include bundles such as vector bundle and line bundle, the response matrix and probabilities, including the cycle-rooted grove probabilities and the Green's function. The formulas using the Green's function are also presented.