The Annals of Probability
- Ann. Probab.
- Volume 39, Number 5 (2011), 1983-2017.
Spanning forests and the vector bundle Laplacian
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.
Ann. Probab., Volume 39, Number 5 (2011), 1983-2017.
First available in Project Euclid: 18 October 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Kenyon, Richard. Spanning forests and the vector bundle Laplacian. Ann. Probab. 39 (2011), no. 5, 1983--2017. doi:10.1214/10-AOP596. https://projecteuclid.org/euclid.aop/1318940787