Parabolic-hyperbolic dichotomy through half-plane coexistence

TL;DR

This paper establishes a connection between the geometric type of unimodular random planar maps and the coexistence of infinite clusters in percolation, revealing a dichotomy based on half-plane coexistence.

Contribution

It extends the parabolic-hyperbolic dichotomy to unimodular random planar maps, linking geometric type to percolation cluster coexistence.

Findings

URM is parabolic iff no half-plane coexistence occurs

URM is hyperbolic iff half-plane coexistence occurs

Generalizes previous results from $\\mathbb{Z}^2$ to URMs

Abstract

Consider a unimodular random planar map (URM) with an invariant ergodic percolation having infinite primal and dual clusters. We say that there is half-plane coexistence if both the percolation and its dual have infinite clusters when restricted to a half-plane. Under mild assumptions on the percolation, we show that the URM is parabolic if and only if there is no half-plane coexistence, and it is hyperbolic if and only if there is half-plane coexistence. This extends the recent half-plane non-coexistence result for $\mathbb{Z}^2$ by Klausen and Kravitz and provides another manifestation of the parabolic-hyperbolic dichotomy for URM's.