## 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 language | English |
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Number of pages | 41 |

Journal | Random Structures and Algorithms |

Early online date | 15 Dec 2017 |

DOIs | |

Publication status | E-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