Random curves on surfaces induced from the Laplacian determinant

Abstract

We define natural probability measures on finite multicurves (finite collections of pairwise disjoint simple closed curves) on curved surfaces. These measures arise as universal scaling limits of probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric, in the limit as the mesh size tends to zero. These in turn are defined from the Laplacian determinant and depend on the choice of a unitary connection on the surface.

Wilson’s algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence.

We set the framework for the study of these probability measures and their scaling limits and state some of their properties.