Hyperbolic and Parabolic Unimodular Random Maps

TL;DR

This paper explores the deep connections between geometric, probabilistic, and combinatorial properties of infinite planar unimodular random maps, showing they are governed by local curvature and establishing their soficity as limits of finite maps.

Contribution

It demonstrates the equivalence of various global properties based on local curvature and proves all simply connected unimodular maps are sofic limits of finite maps.

Findings

Global properties are determined by local average curvature.

Many properties like amenability and percolation are equivalent.

All simply connected unimodular maps are sofic.

Abstract

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.