Geometry of the Uniform Infinite Half-Planar Quadrangulation

TL;DR

This paper introduces a new construction of the uniform infinite half-planar quadrangulation (UIHPQ), analyzes its geometric properties, and reveals its boundary distance process converges to Bessel processes, with implications for geodesic structure and submap decomposition.

Contribution

It provides a novel construction of the UIHPQ, studies its boundary distance process, and explores its geodesic structure and decomposition into independent submaps.

Findings

Boundary distance process converges to Bessel processes of dimension 5.

The UIHPQ can be decomposed into three independent submaps via geodesic pencils.

Large-radius balls are on average 7/9 the size of those in the UIPQ.

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 M\'enard, Miermont and the second author did for the UIPQ, and prove that it induces a decomposition of the UIHPQ into three 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.