scholarly journals Classical Dimers on Penrose Tilings

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Felix Flicker ◽  
Steven H. Simon ◽  
S. A. Parameswaran
Keyword(s):  
Penrose Tilings

Percolation on Penrose Tilings

1998 ◽  
Vol 41 (2) ◽  
pp. 166-177 ◽  
Author(s):  
A. Hof
Keyword(s):  
Large Class ◽  
Arbitrary Dimension ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Aperiodic Tilings ◽  
Infinite Clusters ◽  
Almost All ◽  
Percolation Processes ◽  
Penrose Tilings

AbstractIn 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 ℤd, and to other percolation processes, including Bernoulli bond percolation.

scholarly journals Structural transformations in quasicrystals induced by higher dimensional lattice transitions

2012 ◽  
Vol 468 (2141) ◽  
pp. 1452-1471 ◽  
Author(s):  
Giuliana Indelicato ◽  
Tom Keef ◽  
Paolo Cermelli ◽  
David G. Salthouse ◽  
Reidun Twarock ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Higher Dimensions ◽  
Structural Transformations ◽  
Icosahedral Symmetry ◽  
Local Transformation ◽  
Project Method ◽  
Higher Dimensional ◽  
Transformation Mechanisms ◽  
Transition Paths ◽  
Penrose Tilings

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

LATTICE GAS MODELS ON SELF-SIMILAR APERIODIC TILINGS

1991 ◽  
Vol 03 (02) ◽  
pp. 163-221 ◽  
Author(s):  
C. P. M. GEERSE ◽  
A. HOF
Keyword(s):  
Lattice Gas ◽  
Arbitrary Dimension ◽  
Quantum Systems ◽  
Translation Invariance ◽  
Aperiodic Tilings ◽  
Mean Pressure ◽  
Lattice Gas Models ◽  
Self Similar ◽  
The Mean ◽  
Penrose Tilings

We discuss lattice gas models on the vertices of tilings in arbitrary dimension that are self-similar in the way Penrose tilings of the plane are self-similar. Among these, there are systems that fundamentally lack translation invariance. Under natural hypotheses on the interactions and the states, we prove the existence of thermodynaraic functions — the mean pressure, the mean energy and the mean entropy — and derive the variational principle. The relation between Gibbs states and tangent functionals to the mean pressure is investigated. Generalizations to quantum systems are also discussed. Our work extends results known for lattice gas models on periodic lattices.

scholarly journals C*-Algebras Associated with Mauldin–Williams Graphs

2008 ◽  
Vol 51 (4) ◽  
pp. 545-560 ◽  
Author(s):  
Marius Ionescu ◽  
Yasuo Watatani
Keyword(s):  
Iterated Function System ◽  
Cyclic Permutation ◽  
Invariant Set ◽  
C Algebras ◽  
Open Set ◽  
Iterated Function ◽  
Inflation Rule ◽  
Penrose Tilings ◽  
Williams Graph

AbstractA Mauldin–Williams graph is a generalization of an iterated function system by a directed graph. Its invariant set K plays the role of the self-similar set. We associate a C*-algebra (K) with a Mauldin–Williams graph and the invariant set K, laying emphasis on the singular points. We assume that the underlying graph G has no sinks and no sources. If satisfies the open set condition in K, and G is irreducible and is not a cyclic permutation, then the associated C*-algebra (K) is simple and purely infinite. We calculate the K-groups for some examples including the inflation rule of the Penrose tilings.

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