The Annals of Probability

Determinantal spanning forests on planar graphs

Richard Kenyon

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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.

Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 952-988.

Dates
Received: February 2017
Revised: January 2018
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1551171642

Digital Object Identifier
doi:10.1214/18-AOP1276

Mathematical Reviews number (MathSciNet)
MR3916939

Zentralblatt MATH identifier
07053561

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

Keywords
Graph Laplacian spanning forest determinantal process limit shape

Citation

Kenyon, Richard. Determinantal spanning forests on planar graphs. Ann. Probab. 47 (2019), no. 2, 952--988. doi:10.1214/18-AOP1276. https://projecteuclid.org/euclid.aop/1551171642


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