Percolation Clusters in Hyperbolic Tessellations
نویسنده
چکیده
It is known that, for site percolation on the Cayley graph of a cocompact Fuchsian group of genus ≥ 2, infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = P{site is open}. In such cases, the set Λ of limit points at ∞ of an infinite cluster is a perfect, nowhere dense set of Lebesgue measure 0. In this paper, a variational formula for the Hausdorff dimension δH(Λ) is proved, and used to deduce that δH(Λ) is a continuous, strictly increasing function of p that converges to 0 and 1 at the lower and upper boundaries, respectively, of the coexistence phase.
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