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arXiv · 1708.05876

Percolation Thresholds in Hyperbolic Lattices

Stephan Mertens·Cristopher Moore
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Abstract

We use invasion percolation to compute numerical values for bond and site percolation thresholds $p_c$ (existence of an infinite cluster) and $p_u$ (uniqueness of the infinite cluster) of tesselations $\{P,Q\}$ of the hyperbolic plane, where $Q$ faces meet at each vertex and each face is a $P$-gon. Our values are accurate to six or seven decimal places, allowing us to explore their functional dependency on $P$ and $Q$ and to numerically compute critical exponents. We also prove rigorous upper and lower bounds for $p_c$ and $p_u$ that can be used to find the scaling of both thresholds as a function of $P$ and $Q$.

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BibTeXRIS

Stephan Mertens, Cristopher Moore. 2017-10-10. Percolation Thresholds in Hyperbolic Lattices. https://doi.org/10.1103/physreve.96.042116

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