Geodesic
Uniform infinite planar triangulation and related branching process
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 141-174 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{ZNSL_2004_307_a4,
     author = {M. A. Krikun},
     title = {Uniform infinite planar triangulation and related branching process},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {141--174},
     year = {2004},
     volume = {307},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a4/}
}
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M. A. Krikun. Uniform infinite planar triangulation and related branching process. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 141-174. http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a4/

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