UNIFORM SPANNING FORESTS OF PLANAR GRAPHS

Tom Hutchcroft; Asaf Nachmias

doi:https://doi.org/10.1017/fms.2019.14

UNIFORM SPANNING FORESTS OF PLANAR GRAPHS

Tom Hutchcroft, Asaf Nachmias

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Article number	e29
Journal	Forum of Mathematics, Sigma
Volume	7
DOIs	https://doi.org/10.1017/fms.2019.14
State	Published - 2019

All Science Journal Classification (ASJC) codes

Analysis
Theoretical Computer Science
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Mathematics

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https://doi.org/10.1017/fms.2019.14

Cite this

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abstract = "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.",

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AB - 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.

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JO - Forum of Mathematics, Sigma

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UNIFORM SPANNING FORESTS OF PLANAR GRAPHS

Abstract

All Science Journal Classification (ASJC) codes

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