Monotonicity of two-point connectivity in lattice percolation

Determine whether, for Bernoulli bond percolation on lattice graphs such as the integer lattice ℤ^d (d≥2) at fixed parameter p∈(0,1), the two-point connectivity probability P_p[u v] is monotone in the distance |u−v|, i.e., ascertain whether P_p[u v] is a nonincreasing function of |u−v|.

Background

In lattice percolation, two-point functions P_p[u v] measure connectivity probabilities between vertices as a function of their separation. While significant progress exists for high-dimensional percolation via lace expansion and related techniques, a basic monotonicity-in-distance property remains unsettled.

The paper notes that this monotonicity would follow from the bunkbed conjecture; although the conjecture is disproved here in general, the question for grid-like graphs remains of independent interest. Monotonicity is known in the limit p↓0 and for the Ising model, but not for general percolation.