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UNIFORM SPANNING FORESTS OF PLANAR GRAPHS

Part of: Geometric probability and stochastic geometry Graph theory

Published online by Cambridge University Press:  13 September 2019

TOM HUTCHCROFT  and
ASAF NACHMIAS
TOM HUTCHCROFT
Affiliation:
Statslab, DPMMS, University of Cambridge, Cambridge CB3 0WB, UK; t.hutchcroft@maths.cam.ac.uk
ASAF NACHMIAS
Affiliation:
Department of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel; asafnach@tauex.tau.ac.il

Abstract

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We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e29
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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