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Percolation on Penrose Tilings

Published online by Cambridge University Press:  20 November 2018

A. Hof
A. Hof*
Affiliation:
Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1
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Abstract

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In Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ${{\mathbb{Z}}^{d}}$, and to other percolation processes, including Bernoulli bond percolation.

Keywords

60K3582B43
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 2 , 01 June 1998 , pp. 166 - 177
Copyright
Copyright © Canadian Mathematical Society 1998

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