Summary
We consider a percolation model on the plane which consists of 1-dimensional sticks placed at points of a Poisson process onR 2 each stick having a random, but bounded length and a random direction. The critical probabilities are defined with respect to the occupied clusters and vacant clusters and they are shown to be equal. The equality is shown through a ‘pivotal cell’ argument, using a version of the Russo-Seymour-Welsh theorem which we obtain for this model.
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Roy, R. Percolation of poisson sticks on the plane. Probab. Th. Rel. Fields 89, 503–517 (1991). https://doi.org/10.1007/BF01199791
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DOI: https://doi.org/10.1007/BF01199791