Percolation clusters in hyperbolic tessellations |
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Authors: | SP Lalley |
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Institution: | (1) Department of Statistics, University of Chicago, Chicago, IL 60637-1514, USA, e-mail: lalley@galton.uchicago.edu, US |
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Abstract: | It is known that, for site percolation on the Cayley graph of a co-compact Fuchsian group of genus , infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = P{site is open}. In such cases, the set 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 is proved, and used to deduce that 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.
Submitted: July 2000. |
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Keywords: | |
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