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A new lower bound for the critical probability of site percolation on the square lattice

โœ Scribed by J. van den Berg; A. Ermakov

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
705 KB
Volume
8
Category
Article
ISSN
1042-9832

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โœฆ Synopsis

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 (Toth [13]), 0.522 (Zuev [15]), and the best lower bound so far, 0.541 (Menshikov and Pelikh [12]). 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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## Abstract A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. Let __scs__(__n__) denote the smallest possible size of a critical set in a latin square of order __n__. We show that for all __n__, $scs(n)\geq n\lfloo