PERFECT MATCHING RULES FOR UNDECORATED TRIANGULAR TILINGS WITH 10-, 12-, AND 8-FOLD SYMMETRY

Perfect matching rules are derived for quasiperiodic triangular tilings with 10-, 12-, and 8-fold symmetry. We use the composition/decomposition approach via local inflation/deflation symmetry and emphasize the locality of our procedure: The matching rules given here are formulated using certain decorations of the tilings. These decorations turn out to be redundant, i.e., they are locally derivable from the undecorated tilings. Hence, the latter are determined by perfect matching rules as well.

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Representation of certain self-similar quasiperiodic tilings with perfect matching rules by discrete point sets

Journal de Physique I ◽

10.1051/jp1:1994234 ◽

1994 ◽

Vol 4 (6) ◽

pp. 893-904 ◽

Cited By ~ 4

Author(s):

Richard Klitzing ◽

Michael Baake

Keyword(s):

Perfect Matching ◽

Discrete Point ◽

Point Sets ◽

Self Similar ◽

Matching Rules ◽

Quasiperiodic Tilings

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Translation-Equivariant Matchings of Coin Flips on ℤd

Advances in Applied Probability ◽

10.1017/s0001867800003918 ◽

2010 ◽

Vol 42 (01) ◽

pp. 69-82

Author(s):

Terry Soo

Keyword(s):

Perfect Matching ◽

Fair Coin ◽

Matching Rule ◽

Deterministic Function ◽

Tail Behaviour ◽

Matching Rules

Consider independent fair coin flips at each site of the lattice ℤd. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤdand is given by a deterministic function of the coin flips. LetZΦbe the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour ofZΦfor translation-equivariant perfect matching rules? We prove that, for everyd≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε< ∞ for every ε > 0.

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Translation-Equivariant Matchings of Coin Flips on ℤd

Advances in Applied Probability ◽

10.1239/aap/1269611144 ◽

2010 ◽

Vol 42 (1) ◽

pp. 69-82 ◽

Cited By ~ 1

Author(s):

Terry Soo

Keyword(s):

Perfect Matching ◽

Fair Coin ◽

Matching Rule ◽

Deterministic Function ◽

Tail Behaviour ◽

Matching Rules

Consider independent fair coin flips at each site of the lattice ℤd. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤd and is given by a deterministic function of the coin flips. Let ZΦ be the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour of ZΦ for translation-equivariant perfect matching rules? We prove that, for every d ≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε < ∞ for every ε > 0.

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