Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary

Erich Baur (1) , Grégory Miermont (1) , Loïc Richier (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 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.

Mots clés: Uniform infinite half-planar quadrangulation; geodesic rays; boundary

Domaines: Probabilités [math.PR]

Erich Baur, Grégory Miermont, Loïc Richier. Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2016, XIII (2), pp.1123-1149. ⟨hal-01316514v2⟩