A combinatorial criterion for macroscopic circles in planar triangulations
Probability
2019-06-05 v1 Metric Geometry
Abstract
Given a finite simple triangulation, we estimate the sizes of circles in its circle packing in terms of Cannon's vertex extremal length. Our estimates provide control over the size of the largest circle in the packing. We use them, combined with results from [12], to prove that in a proper circle packing of the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield, the size of the largest circle goes to zero with high probability.
Keywords
Cite
@article{arxiv.1906.01612,
title = {A combinatorial criterion for macroscopic circles in planar triangulations},
author = {Ori Gurel-Gurevich and Daniel C. Jerison and Asaf Nachmias},
journal= {arXiv preprint arXiv:1906.01612},
year = {2019}
}
Comments
13 pages, 3 figures