New bounds for the site percolation threshold of the hexagonal lattice

Abstract

The site percolation threshold of the hexagonal lattice satisfies $0.656246 < p_c < 0.739695$. For comparison, the largest previous lower bound of 0.652703... was established in 1981, and the smallest previous upper bound of 0.743359 was derived in 2007. The bound is obtained by using the substitution method to compare the hexagonal lattice site model to an exactly-solved two-parameter site percolation model on the martini lattice. Computational reductions involving graph-welding, symmetry, non-crossing partitions, and network flow computations overcome challenges to establishing stochastic ordering between the models.