Alignment Percolation

Abstract

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in \(d \geqslant 2\) dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on \(\mathbb {Z}^{d}\) with parameter p ∈ (0,1]. For each occupied site v, and for each of the 2d 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 2d incident segments to be blue. In the ‘independent model’, the states of different line segments are independent.

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Acknowledgements

NRB is supported by grant DE170100186 from the Australian Research Council. MH is supported by Future Fellowship FT160100166 from the Australian Research Council. The authors are grateful to a colleague and two referees for their comments.

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Correspondence to Geoffrey R. Grimmett.

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Beaton, N.R., Grimmett, G.R. & Holmes, M. Alignment Percolation. Math Phys Anal Geom 24, 3 (2021). https://doi.org/10.1007/s11040-021-09373-7

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Keywords

  • Percolation
  • One-choice model
  • Independent model

Mathematics Subject Classification (2010)

  • 60K35