publication . Preprint . Article . Other literature type . 2019

Determinantal spanning forests on planar graphs

Richard Kenyon;
Open Access English
  • Published: 01 Mar 2019
Abstract
We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures arise naturally from the Laplacian on the graph. ¶ More generally, these results hold for the “massive” Laplacian determinant which counts rooted spanning forests with weight $M$ per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is always a simple Harnack curve; this fact controls the decay of edge-edg...
Subjects
free text keywords: Mathematics - Probability, 82B20, 60Cxx, Statistics, Probability and Uncertainty, Statistics and Probability, Graph Laplacian, spanning forest, determinantal process, limit shape, 82B20, Minimum spanning tree, Discrete mathematics, Spanning tree, Euclidean minimum spanning tree, Connected dominating set, Loop-erased random walk, Combinatorics, k-minimum spanning tree, Minimum degree spanning tree, Mathematics, Laplacian matrix
Related Organizations
Funded by
NSF| Integrability and limit shapes in the two-dimensional Ising model and related models
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1208191
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences

[2] I. Benjamini, R. Lyons. Y. Peres, O. Schramm, Uniform spanning forests. Ann. Probab. 29 (2001), no. 1, 165

[10] R. Kenyon, A. Okounkov, Planar dimers and Harnack curves Duke Math. J. 131 (2006), no. 3, 499{524.

[11] R. Kenyon, A. Okounkov, Limit shapes and the complex Burgers equation. Acta Math. 199 (2007), no. 2, 263302.

[12] R. Kenyon, A. Okounkov, S. She eld, Dimers and amoebae. Ann. of Math. (2) 163 (2006), no. 3, 10191056.

[13] R. Kenyon, J. Propp, D. Wilson Trees and matchings. Electron. J. Combin. 7 (2000), Research Paper 25, 34 pp.

[15] R. Kenyon and D. Wilson, Boundary partitions in trees and dimers. Trans. Amer. Math. Soc. 363 (2011), no. 3, 13251364

[16] G. Lawler, O. Schramm, W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004), no. 1B, 939995.

[17] R. Lyons, Determinantal probability measures, Publ. Math. Inst. Hautes Etudes Sci. 98 (2003), 167{212.

[25] Wangru Sun, Toroidal Dimer Model and Temperley's Bijection, arxiv:1603.00690 (2016)

[26] H. N. V. Temperley. In Combinatorics: Proceedings of the British Combinatorial Conference 1973, 13:202 204, 1974.

[27] D. Wilson, Generating random spanning trees more quickly than the cover time. Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), 296303, ACM, New York, 1996.

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