The Annals of Probability

Spanning forests and the vector bundle Laplacian

Richard Kenyon

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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.

Article information

Ann. Probab., Volume 39, Number 5 (2011), 1983-2017.

First available in Project Euclid: 18 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Discrete Laplacian quaternion spanning tree resistor network


Kenyon, Richard. Spanning forests and the vector bundle Laplacian. Ann. Probab. 39 (2011), no. 5, 1983--2017. doi:10.1214/10-AOP596.

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