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.

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.

Upper bounds for the critical probability of oriented percolation in two dimensions

1993 ◽  
Vol 440 (1908) ◽  
pp. 201-220 ◽  
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation

We give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions. This method enables us to prove that the critical probability is at most 0.6863, greatly improving the best published upper bound, 0.84. We also prove that our method can be used to give arbitrarily good upper bounds. We also use a slight variant of our method to obtain an upper bound of 0.72599 for the critical probability in oriented site percolation.

On spàtial general epidemics and bond percolation processes

1984 ◽  
Vol 21 (4) ◽  
pp. 911-914 ◽  
Author(s):  
Kari Kuulasmaa ◽  
Stan Zachary
Keyword(s):  
Lower Bound ◽  
Bond Percolation ◽  
Percolation Process ◽  
Percolation Probability ◽  
Percolation Processes

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.

On spàtial general epidemics and bond percolation processes

1984 ◽  
Vol 21 (04) ◽  
pp. 911-914 ◽  
Author(s):  
Kari Kuulasmaa ◽  
Stan Zachary
Keyword(s):  
Lower Bound ◽  
Bond Percolation ◽  
Percolation Process ◽  
Percolation Probability ◽  
Percolation Processes

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.

scholarly journals New Riemannian manifolds with Lp-unbounded Riesz transform for p > 2

2020 ◽  
Author(s):  
Alex Amenta
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Fractal Structure ◽  
Arbitrary Dimension ◽  
Riesz Transform

Abstract We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $$L^p(M)$$ L p ( M ) for all $$p > 2$$ p > 2 . This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.

Bratteli diagrams via the De Concini–Procesi theorem

2020 ◽  
pp. 2050073
Author(s):  
Daniele Mundici
Keyword(s):  
Large Class ◽  
Word Problem ◽  
Bratteli Diagram ◽  
Von Neumann ◽  
Generators And Relations ◽  
The Core ◽  
Bratteli Diagrams ◽  
Group Presentations ◽  
Almost All ◽  
Af Algebras

An AF algebra [Formula: see text] is said to be an AF[Formula: see text] algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF[Formula: see text] algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF[Formula: see text] algebra [Formula: see text], generates a Bratteli diagram of [Formula: see text] We generalize this result to the case when [Formula: see text] has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

On the time constant and path length of first-passage percolation

1980 ◽  
Vol 12 (04) ◽  
pp. 848-863 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Distribution Function ◽  
Time Constant ◽  
Path Length ◽  
Passage Time ◽  
Square Lattice ◽  
Bond Percolation ◽  
First Passage Percolation ◽  
Critical Probability ◽  
First Passage ◽  
Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT , then there exist constants 0 < a, C 1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C 1 n). From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn (c) <∞, where Nn (c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

scholarly journals Percolation of Words on Zd with Long-Range Connections

2011 ◽  
Vol 48 (4) ◽  
pp. 1152-1162 ◽  
Author(s):  
B. N. B. de Lima ◽  
R. Sanchis ◽  
R. W. C. Silva
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Binary Sequences ◽  
Site Percolation ◽  
Almost All ◽  
Independent Site

Consider an independent site percolation model on Zd, with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1}N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

scholarly journals Line-of-Sight Percolation

2009 ◽  
Vol 18 (1-2) ◽  
pp. 83-106 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
SVANTE JANSON ◽  
OLIVER RIORDAN
Keyword(s):  
Random Walk ◽  
Higher Dimensions ◽  
Percolation Model ◽  
The Other ◽  
Branching Random Walk ◽  
Line Of Sight ◽  
Continuum Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Vertex Set

Given ω ≥ 1, let $\Z^2_{(\omega)}$ be the graph with vertex set $\Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus $\Z^2_{(1)}$ is precisely $\Z^2$.) Let pc(ω) be the critical probability for site percolation on $\Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

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