Spanning forests and the vector bundle Laplacian

TL;DR

This paper extends the classical matrix-tree theorem to Laplacians on vector bundles, linking their determinants to cycle rooted spanning forests and revealing new probabilistic properties of loop-erased random walks.

Contribution

It introduces a generalized combinatorial interpretation of Laplacian determinants for vector bundles and constructs determinantal measures on cycle rooted spanning forests.

Findings

Determinantal process for edges in CRSFs

Generalization of spanning tree process to surface-embedded graphs

Calculation of loop-erased random walk probabilities using the new framework

Abstract

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.