Percolation clusters in hyperbolic tessellations

Abstract

It is known that, for site percolation on the Cayley graph of a co-compact Fuchsian group of genus $ \ge 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 $ \Lambda $ of limit points at $ \infty $ of an infinite cluster is a perfect, nowhere dense set of Lebesgue measure 0. In this paper, a variational formula for the Hausdorff dimension $ \delta_H(\Lambda) $ is proved, and used to deduce that $ \delta_H(\Lambda) $ 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.