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.