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.
In REF Laplacians on one-and two-dimensional vector bundles on graphs were used to understand the relation between graphs embedded on surfaces and cycle rooted spanning forests, generalizing the relation between spanning trees and graph Laplacians.
Spanning forests and the vector bundle Laplacian
Annals of Probability 2011, Vol. 39, No. 5, 1983-2017
Mathematics Physics