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March 2017 Random curves on surfaces induced from the Laplacian determinant
Adrien Kassel, Richard Kenyon
Ann. Probab. 45(2): 932-964 (March 2017). DOI: 10.1214/15-AOP1078

Abstract

We define natural probability measures on finite multicurves (finite collections of pairwise disjoint simple closed curves) on curved surfaces. These measures arise as universal scaling limits of probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric, in the limit as the mesh size tends to zero. These in turn are defined from the Laplacian determinant and depend on the choice of a unitary connection on the surface.

Wilson’s algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence.

We set the framework for the study of these probability measures and their scaling limits and state some of their properties.

Citation

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Adrien Kassel. Richard Kenyon. "Random curves on surfaces induced from the Laplacian determinant." Ann. Probab. 45 (2) 932 - 964, March 2017. https://doi.org/10.1214/15-AOP1078

Information

Received: 1 October 2014; Revised: 1 October 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 1377.82037
MathSciNet: MR3630290
Digital Object Identifier: 10.1214/15-AOP1078

Subjects:
Primary: 82B20

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 2 • March 2017
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