Abstract
We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. We also prove a finite analogue of this statement, valid for expander graphs, without any girth assumption.
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Benjamini, I., Nachmias, A. & Peres, Y. Is the critical percolation probability local?. Probab. Theory Relat. Fields 149, 261–269 (2011). https://doi.org/10.1007/s00440-009-0251-5
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DOI: https://doi.org/10.1007/s00440-009-0251-5