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Corner percolation on \(\mathbb Z^{2}\) and the square root of 17. (English) Zbl 1159.60032

The author refers to the corner percolation model introduced by Bálint Tóth. Losely speaking, it is a bond percolation model on \(\mathbb Z^2\) in which every edge is present with probability \(\frac12\) and each vertex has exactly two incident edges, perpendicular to each other. Two main results are stated about this process. Firstly, it is shown that each vertex is enclosed by infinitely many cycles, and the expected diameter of the cycle containing the origin is infinite. Secondly, the author determines the value of the tail probability exponent. The mathematical apparatus is related to Markov chain, an the conclusion displays open problems.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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