Consider two dimensional anisotropic Bernoulli percolation where the parameter in the vertical direction, $p_v$, is strictly larger than the parameter in the horizontal direction, $p_h$. Suppose that $(x_1, x_2)$ satisfies $x_1<x_2$. A number of papers have addressed the question of under what restrictions we have that the two point function $P_{(p_h, p_v)}(0\leftrightarrow (x_1, x_2))>P_{(p_h, p_v)}(0\leftrightarrow (x_2, x_1)).$ This gives me the impression that we always have the nonstrict inequality "$\ge$", but I wanted to check to see if this was in agreement with the reality of what we know. I cannot find any literature supporting the hypothesis that nonstrict inequality always holds.
I ask the same question for the finite connectivity function as well.
My question makes no reference to critical phenomena. But does current research seem to indicate that we should define a critical surface in $(p_h, p_v)$ and then ask the above question only above (or below, or extremely above or extremely below) the critical surface?