Geometry of the uniform infinite half-planar quadrangulation

Alessandra Caraceni, Nicolas Curien

Research output: Contribution to journalArticlepeer-review

6 Citations (SciVal)

Abstract

We give a new construction of the uniform infinite half-planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus, which allows us to perform a detailed study of its geometry. We show that the process of distances to the root vertex read along the boundary contour of the UIHPQ evolves as a particularly simple Markov chain and converges to a pair of independent Bessel processes of dimension 5 in the scaling limit. We study the "pencil" of infinite geodesics issued from the root vertex as reported by Curien, Ménard, and Miermont and prove that it induces a decomposition of the UIHPQ into 3 independent submaps. We are also able to prove that balls of large radius around the root are on average 7/9 times as large as those in the UIPQ, both in the UIHPQ and in the UIHPQ with a simple boundary; this fact we use in a companion paper to study self-avoiding walks on large quadrangulations.

Original languageEnglish
Number of pages41
JournalRandom Structures and Algorithms
Early online date15 Dec 2017
DOIs
Publication statusE-pub ahead of print - 15 Dec 2017

Keywords

  • Brownian plane
  • Geodesic rays
  • Pencil decomposition
  • Random planar maps
  • Self-avoiding walk
  • Uniform Infinite Half-Planar Quadrangulation

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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