Uniform Spanning Forests of Planar Graphs

TL;DR

This paper proves the connectivity of free uniform spanning forests in bounded degree proper plane graphs and explores their geometric properties using hyperbolic geometry, revealing universal critical exponents.

Contribution

It establishes the almost sure connectivity of uniform spanning forests in bounded degree plane graphs and introduces a hyperbolic geometric framework for analyzing their critical exponents.

Findings

Uniform spanning forests are connected almost surely in bounded degree plane graphs.

Critical exponents governing the geometry are calculated explicitly.

Universal exponents are identified via hyperbolic circle packing geometry.

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.