Monotonicity of Percolation Probability on \Z^d with Long Range Connections

2010/06/21 by de Lima, Bernardo N. B., Sanchis, Rémy, Silva, Roger W. C.
#60 CXX #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1006.4191

Abstract

Consider an independent site percolation model on \Z^d, d≥ 2, with parameter p ∈ (0,1), where there are only nearest neighbor bonds and long range bonds of length k parallel to some coordinate axis. We show that the percolation probability is a non-decreasing function of k. We also generalize this result for models whose long range bonds have several lengths.

Related

• Critical Point and Percolation Probability in a Long Range Site Percolation Model on \Z^d
• Is the Percolation Probability on ℤ^d with Long Range Connections Monotone?
• Distances in critical long range percolation
• Percolation of words on \Z^d with long range connections
• Monotonicity and phase diagram for multi-range percolation on oriented trees