Coordination sequences for hyperbolic tilings

O'keefe, M.; Keeffe,

doi:10.1524/zkri.1998.213.3.135

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Appendix A: Counting Vertices In Hyperbolic Lattices1

Introduction1

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2003

2024

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Cited by 5 publications

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“…Let n k denote the number of vertices in the hyperbolic lattice with graph distance k from the origin. This number is again given by a linear recurrence, albeit a more complicated one, which was derived independently in physics [36] and in mathematics [37]. We review this here, using the notation of [36].…”

Section: Appendix A: Counting Vertices In Hyperbolic Latticesmentioning

confidence: 99%

Percolation thresholds in hyperbolic lattices

Mertens

Moore

2017

Phys. Rev. E

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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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Section: Appendix A: Counting Vertices In Hyperbolic Latticesmentioning

confidence: 99%

Percolation thresholds in hyperbolic lattices

Mertens

Moore

2017

Phys. Rev. E

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“…The coordination sequences of periodic graphs are predicted to be of quasi-polynomial type (see Definition 1.2) by Grosse-Kunstleve et al (1996). After that, various mathematical methods to calculate coordination sequences have been developed and they are actually calculated in many specific cases as in the work of Conway & Sloane (1997), Eon (2002Eon ( , 2012, Goodman-Strauss & Sloane (2019), O'Keeffe (1995O'Keeffe ( , 1998, Shutov & Maleev (2018, 2020.…”

Section: Introductionmentioning

confidence: 99%