The Annals of Probability

Spanning forests and the vector bundle Laplacian

Richard Kenyon

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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.

Article information

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

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1318940787

Digital Object Identifier
doi:10.1214/10-AOP596

Mathematical Reviews number (MathSciNet)
MR2884879

Zentralblatt MATH identifier
1252.82029

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

Keywords
Discrete Laplacian quaternion spanning tree resistor network

Citation

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


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References

  • [1] Boutillier, C. and de Tilière, B. (2009). Loop statistics in the toroidal honeycomb dimer model. Ann. Probab. 37 1747–1777.
  • [2] Brooks, R. L., Smith, C. A. B., Stone, A. H. and Tutte, W. T. (1940). The dissection of rectangles into squares. Duke Math. J. 7 312–340.
  • [3] Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 1329–1371.
  • [4] Fock, V. and Goncharov, A. (2006). Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103 1–211.
  • [5] Forman, R. (1993). Determinants of Laplacians on graphs. Topology 32 35–46.
  • [6] Goncharov, A. and Kenyon, R. (2010). Dimers and cluster varieties. Unpublished manuscript, Brown Univ.
  • [7] Hammond, A. and Kenyon, R. (2011). Monotone loops models and rational resonance. Probab. Theory Related Fields 150 613–633.
  • [8] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Probab. Surv. 3 206–229 (electronic).
  • [9] Kenyon, R. (2010). Loops in the double-dimer model. Unpublished manuscript, Brown Univ.
  • [10] Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239–286.
  • [11] Kenyon, R. (2000). Long-range properties of spanning trees. J. Math. Phys. 41 1338–1363.
  • [12] Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128–1137.
  • [13] Kenyon, R. and Wilson, D. (2010). Laplacians on surface graphs. Unpublished manuscript, Brown Univ.
  • [14] Kenyon, R. W. and Wilson, D. B. (2004). Critical resonance in the non-intersecting lattice path model. Probab. Theory Related Fields 130 289–318.
  • [15] Kirchhoff, F. (1847). Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72 497–508.
  • [16] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
  • [17] Majumdar, S. (1992). Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions. Phys. Rev. Lett. 68 2329–2331.
  • [18] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier, Amsterdam.
  • [19] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574.
  • [20] Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) 296–303. ACM, New York.