Alignment Percolation

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in d ≥ 2 dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ℤ d with parameter p ∈ (0,1]. For each occupied site v...

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Bibliographic Details
Published inMathematical physics, analysis, and geometry Vol. 24; no. 1
Main Authors Beaton, Nicholas R., Grimmett, Geoffrey R., Holmes, Mark
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.03.2021
Springer Nature B.V
Subjects
Analysis
Applications of Mathematics
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Percolation
Phase diagrams
Physics
Physics and Astronomy
Segments
Stochastic models
Theoretical
Two dimensional models
One-choice model
Percolation
60K35
Independent model
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Summary:The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in d ≥ 2 dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ℤ d with parameter p ∈ (0,1]. For each occupied site v , and for each of the 2 d possible coordinate directions, declare the entire line segment from v to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the ‘one-choice model’, each occupied site declares one of its 2 d incident segments to be blue. In the ‘independent model’, the states of different line segments are independent.
ISSN:1385-0172
1572-9656
DOI:10.1007/s11040-021-09373-7