A remark on monotonicity in Bernoulli bond Percolation

Abstract: Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\geq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}({(0,\dots,0)\leftrightarrow (n,0,\dots,0)})$ is a monotone function in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}({(0,\dots,0)\leftrightarrow (n,0,\dots,0), (0,\dots,0)\nleftrightarrow\infty})$ is also a monotone function in $n$ when $p$ is close to 1.