Geometry of the Uniform Infinite Half-Planar Quadrangulation

Understand

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.