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Percolation on random triangulations and stable looptrees. (English) Zbl 1342.60164

The study is devoted to site percolation on the uniform infinite planar triangulation (UIPT), which was introduced by O. Angel and O. Schramm [Commun. Math. Phys. 241, No.  2–3, 191–213 (2003; Zbl 1098.60010)]. A triangulation is defined as an embedding of a finite connected graph in the two-dimensional sphere such that all faces are triangles, i.e., have degree three. It is assumed that an oriented edge, called the root edge, is distinguished and that loops and multiple edges are allowed. \(T_n\) denotes a triangulation chosen uniformly at random from the set of all rooted triangulations with \(n\) vertices. Under a given UIPT, a site percolation is obtained by coloring vertices independently white with probability \(\alpha\in (0; 1)\) and black with probability \(1-\alpha\). It is known that the critical threshold is \(\alpha_c=1/2\). The main result of the paper concerns the geometry of the boundary of percolation clusters and it says that for \(\alpha=\alpha_c=1/2\), \[ \mathcal P(\#\partial\mathcal H_\alpha^o=n)\sim\frac{3}{2| \Gamma(-2/3)|^3}n^{-4/3},\quad n\to\infty, \] where \(\#\partial\mathcal H_\alpha^o\) denotes the perimeter of the boundary of the white hull \(\partial\mathcal H_\alpha^o\) for a given \(\alpha\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
05C80 Random graphs (graph-theoretic aspects)

Citations:

Zbl 1098.60010
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References:

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