[Paper Review] Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary

Erich Baur, Grégory Miermont+1 more|arXiv (Cornell University)|May 15, 2016

Computational Geometry and Mesh Generation18 references4 citations

TL;DR

This paper proves that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely often, using a Schaeffer-type encoding of the UIHPQ via a two-sided simple random walk and critical Galton-Watson trees. The key result is that geodesic rays are proper and return to the boundary with logarithmic tail distribution of inter-hit times, and the same behavior extends to the uniform infinite half-planar triangulation (UIHPT).

ABSTRACT

We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien (2015) by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.

Motivation & Objective

• To resolve a recent question by Curien on whether geodesic rays in the UIHPQ intersect the boundary infinitely often.
• To establish that geodesic rays in the UIHPQ are proper, extending a result by Caraceni and Curien via a different method.
• To demonstrate that the intersection behavior of geodesic rays with the boundary is quantitatively similar in both the UIHPQ and UIHPT.
• To extend results from the quadrangulation model to the triangulation model using a variant of the Schaeffer-type encoding.
• To show that the set of boundary vertices visited by all geodesic rays is infinite and coarsely distributed, with explicit tail behavior for inter-hit times.

Proposed method

• Uses a Schaeffer-type encoding of the UIHPQ based on a two-sided simple random walk and uniformly labeled critical Galton-Watson trees attached to down-steps.
• Analyzes the minimal label attained in trees attached to excursions above -1 in the random walk to derive the distribution of boundary intersection points.
• Constructs two distinguished geodesics (maximal and minimal) from the root to define a universal set of boundary vertices visited by all geodesic rays.
• Applies the same encoding framework to the uniform infinite half-planar triangulation (UIHPT), using a variant of the mobile bridge construction.
• Employs exact distributional calculations for the hitting times of boundary vertices, particularly computing the tail behavior of inter-hit times.
• Uses symmetry and inclusion arguments to compare the sets of boundary vertices visited by different geodesic rays and to derive bounds on visitation frequency.

Experimental results

Research questions

• RQ1 Do all geodesic rays in the UIHPQ intersect the boundary infinitely many times?
• RQ2 What is the distribution of the times between consecutive boundary intersections for geodesic rays in the UIHPQ?
• RQ3 How does the behavior of geodesic rays in the UIHPQ compare to that in the UIHPT in terms of boundary intersections?
• RQ4 Can the same method used for the UIHPQ be extended to the UIHPT to establish similar intersection properties?
• RQ5 What is the precise law of the set of boundary vertices visited by all geodesic rays in the UIHPQ and UIHPT?

Key findings

• Almost surely, every geodesic ray in the UIHPQ intersects the boundary infinitely many times.
• The set of boundary vertices visited by all geodesic rays contains an infinite sequence of distinct points, with all rays passing through all but finitely many of them.
• The distance between consecutive boundary hits for the maximal geodesic has a tail probability satisfying $\mathbb{P}(\delta > m) \sim 1/\ln m$ as $m \to \infty$.
• For the minimal geodesic, the tail behavior is $\mathbb{P}(\delta' > m) \sim 1/(2\ln m)$, indicating slightly sparser visits.
• The same intersection behavior holds in the UIHPT: all geodesic rays hit the boundary infinitely often, with comparable tail distributions.
• The sets $\tilde{\mathcal{R}}_+ \cup \tilde{\mathcal{R}}_-^{\textup{min}}$ and $\tilde{\mathcal{R}}_+^{\textup{min}} \cup \tilde{\mathcal{R}}_-$ serve as tight bounds for the set of boundary visitation times in the UIHPT.