Spanning forests and the vector bundle Laplacian

Preprint, Other literature type English OPEN
Kenyon, Richard;
  • Publisher: The Institute of Mathematical Statistics
  • Journal: issn: 0091-1798
  • Publisher copyright policies & self-archiving
  • Related identifiers: doi: 10.1214/10-AOP596
  • Subject: resistor network | Mathematical Physics | quaternion | Mathematics - Probability | spanning tree | 82B20 | Discrete Laplacian
    acm: MathematicsofComputing_DISCRETEMATHEMATICS

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 determ... View more
  • References (20)
    20 references, page 1 of 2

    [1] Boutillier, C. and de Tili`ere, B. (2009). Loop statistics in the toroidal honeycomb dimer model. Ann. Probab. 37 1747-1777. MR2561433

    [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. MR0003040

    [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. MR1235419

    [4] Fock, V. and Goncharov, A. (2006). Moduli spaces of local systems and higher Teichmu¨ller theory. Publ. Math. Inst. Hautes E´tudes Sci. 103 1-211. MR2233852

    [5] Forman, R. (1993). Determinants of Laplacians on graphs. Topology 32 35-46. MR1204404

    [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 Unpublished manuscript, Brown Univ.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 Vira´g, B. (2006). Determinantal processes and independence. Probab. Surv. 3 206-229 (electronic). MR2216966

    [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 Unpublished manuscript, Brown Univ.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. MR1819995

  • Metrics
Share - Bookmark