Open Access
September 2011 Spanning forests and the vector bundle Laplacian
Richard Kenyon
Ann. Probab. 39(5): 1983-2017 (September 2011). DOI: 10.1214/10-AOP596


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.


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Richard Kenyon. "Spanning forests and the vector bundle Laplacian." Ann. Probab. 39 (5) 1983 - 2017, September 2011.


Published: September 2011
First available in Project Euclid: 18 October 2011

zbMATH: 1252.82029
MathSciNet: MR2884879
Digital Object Identifier: 10.1214/10-AOP596

Primary: 82B20

Keywords: discrete Laplacian , quaternion , Resistor network , spanning tree

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 5 • September 2011
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