Nicholas R Beaton, Geoffrey R Grimmett, Mark Holmes
MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY | SPRINGER | Published : 2021
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 ℤd 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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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.