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Parabolic-hyperbolic dichotomy through half-plane coexistence

Ádám Timár

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.

Parabolic-hyperbolic dichotomy through half-plane coexistence

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 by Klausen and Kravitz and provides another manifestation of the parabolic-hyperbolic dichotomy for URM's.
Paper Structure (2 sections, 7 theorems, 1 figure)

This paper contains 2 sections, 7 theorems, 1 figure.

Table of Contents

  1. Definitions and preliminaries
  2. Proofs

Key Result

Theorem 1

Suppose $G$ is a URM with dual. Let $\omega$ be a random subgraph of $G$ given by an ergodic invariant edge percolation that satisfies (A) or (B). Suppose further that both $\omega$ and $\omega^\star$ have some infinite cluster. Then (1) $G$ is parabolic if and only if there is no half-plane coexist

Figures (1)

  • Figure 1: Subcases 1 and 2.

Theorems & Definitions (12)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 2 more