Let be a sequence of i.i.d. positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in the ith vertical column to another in the th vertical column by an edge having length . Then declare independently each edge e in the resulting lattice open with probability where and is the length of e. We relate the occurrence of a nontrivial phase transition for this model to moment properties of . More precisely, we prove that the model undergoes a nontrivial phase transition when , for some . On the other hand, when , percolation never occurs for . We also show that the probability of the one-arm event decays no faster than a polynomial in an open interval of parameters p close to the critical point.
Ann. Appl. Probab.
33(4):
3145-3168
(August 2023).
DOI: 10.1214/22-AAP1887
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