Geodesic rays in the uniform infinite half-planar quadrangulation return   to the boundary
  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. 
Erich Baur, Gr\'egory Miermont, Lo\"ic Richier
math.PR