Random curves on surfaces induced from the Laplacian determinant

TL;DR

This paper introduces probability measures on cycle-rooted spanning forests on surfaces derived from the Laplacian determinant, demonstrating their convergence and independence from graph approximations, and develops algorithms for sampling these measures.

Contribution

It defines new measures on CRSFs from the Laplacian determinant, proves their convergence on surfaces, and extends Wilson's algorithm for sampling these measures.

Findings

Measures converge for conformally approximating graphs

Sampling relates to loop-erased random walk

Measures are independent of graph sequence chosen

Abstract

We define natural probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric. These measures arise from the Laplacian determinant and depend on the choice of a unitary connection on the tangent bundle to the surface. We show that, for a sequence of graphs $(G_n)$ conformally approximating the surface, the measures on CRSFs of $G_n$ converge and give a limiting probability measure on finite multicurves (finite collections of pairwise disjoint simple closed curves) on the surface, independent of the approximating sequence. 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.