2018/11/05 by Bock, Bounghun, Damron, Michael, Newman, C. M. +1
#60K35 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1811.01678
In independent bond percolation on ℤd with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for d ≥ 19, the set of such p contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d ≥ 11. We improve this result by showing that for d ≥ 10 and some p>pc, there are infinite paths consisting of "shielded" vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of pc, this bound can be reduced to d ≥ 7. Our methods are elementary and do not require the triangle condition.