Uniform Infinite Planar Triangulation and Related Time-Reversed Critical Branching Process

Abstract

We consider the uniform infinite planar triangulation, which is defined as the weak limit of the uniform distributions on finite triangulations with N triangles as N → ∞. Take the ball of radius R in an infinite triangulation. One of the components of its boundary separates this ball from the infinite part of the triangulation, and we denote its length by ℓ(R).

The main question we study is the asymptotic behavior of the sequence ℓ(R), R = 1, 2,..., called the triangulation profile. First, we prove that the ratio ℓ(R)/R² converges to a nondegenerate random variable. Second, we establish a connection between the triangulation profile and a certain time-reversed critical branching process. Finally, we show that there exists a contour of length linear in R that lies outside of the R-ball and separates this ball from the infinite part of the triangulation. Bibliography: 8 titles.