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March 2019 Determinantal spanning forests on planar graphs
Richard Kenyon
Ann. Probab. 47(2): 952-988 (March 2019). DOI: 10.1214/18-AOP1276

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-edge correlations in these models.

We construct a limit shape theory in these settings, where the limit shapes are defined by measured foliations of fixed isotopy type.

Citation

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Richard Kenyon. "Determinantal spanning forests on planar graphs." Ann. Probab. 47 (2) 952 - 988, March 2019. https://doi.org/10.1214/18-AOP1276

Information

Received: 1 February 2017; Revised: 1 January 2018; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053561
MathSciNet: MR3916939
Digital Object Identifier: 10.1214/18-AOP1276

Subjects:
Primary: 82B20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 2 • March 2019
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