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?



Reflection "monotonicity" of two point function percolation
Last I heard, even the non-strict inequality seemed to be open in the general case. And it is a pretty frustrating situation, because it "should be obvious" ...


The critical surface is known (it is given by $p\_h+p\_v=1$ by a duality argument), but indeed the inequality should hold as soon as $p\_v \geq p\_h$, irrespective of whether we are above or below criticality.


One direction in which it might be done is to look below the critical surface, and far enough from the origin (in other words, in the regime of exponential decay, showing that the exponential rates of decay in the two lattice direction are different and ordered in the natural way). But I am not sure whether even that is known.


At criticality, you can have a look at the recent papers by Grimmett and Manolescu, maybe some cases can be extracted from that as well, but I don't exactly remember if they mention it there ...


