Uniform infinite planar triangulation and related branching process

M. A. Krikun

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 141-174

Résumé

We consider the uniform infinite planar triangulation, which is the weak limit of the uniform distributions on finite rooted sphere triangulations with a given number of triangles $N$ as $N\to\infty$. The main question we study is the asymptotic behaviour of the triangulation profile, which we define as follows. Take a ball of radius $R$ in an infinite triangulation. One of its boundary components separates this ball from the infinite part of the triangulation. We denote the length of this component by $\ell(R)$ and call the sequence $\ell(R)$, $R=1,2,\dots$, the triangulation profile. We prove that the ratio $\ell(R)/R^2$ converges to a nondegenerate random variable. We establish a connection between the triangulation profile and a certain time-reversed critical branching process. We also show that there exists a contour of length linear in $R$ that lies outside the $R$-ball and separates the $R$-ball from the infinite part of the triangulation.