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On the truncated anisotropic long-range percolation on Z2:

Sidoravicius, V ; Surgailis, D ; Vares, M.E
Stochastic Processes and their Applications.  81 (1999)  2 - p. 337-349 , 1999
Link: https://doi.org/10.1016/s0304-4149(98)00107-0