Uniform spanning forests of planar graphs
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.
Additional Information© The Author(s) 2019. 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. Received 24 January 2018; accepted 3 March 2019. TH thanks Tel Aviv University and both authors thank the Issac Newton Institute, where part of this work was carried out, for their hospitality. We also thank Tyler Helmuth for finding several typos in an earlier version of this manuscript. Images of circle packings were created with Ken Stephenson's CirclePack software . This project was supported by ERC starting grant RADNGEOM 676970.
Published - uniform-spanning-forests-of-planar-graphs.pdf
Submitted - 1603.07320.pdf