The critical probability for site percolation on the square lattice is not known exactly. Several authors have given rigorous upper and lower bounds. Some recent lower bounds are (each displayed here with the first three digits) 0.503 [Tóth (1985)], 0.522 [Zuev (1988)] and, the best lower bound so far, 0.541 [Menshikov and Pelikh (1989)]. By a modification of the method of Menshikov and Pelikh we get a significant improvement, namely 0.556. Apart from a few classical results on percolation and coupling, which are explicitly stated in the Introduction, this paper is self-contained.

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van den Berg, J, & Ermakov, A. B. (1995). A new lower bound for the critical probability of site percolation on the square lattice. Department of Operations Research, Statistics, and System Theory [BS]. CWI.