The Annals of Probability

Determinantal spanning forests on planar graphs

Richard Kenyon

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Graph Laplacian spanning forest determinantal process limit shape


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

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