A remark on monotonicity in Bernoulli bond Percolation

TL;DR

This paper investigates monotonicity properties of connectivity functions in anisotropic Bernoulli bond percolation on hypercubic lattices, showing monotonicity near the extreme values of the percolation parameter p.

Contribution

It establishes that certain connectivity functions are monotone in distance n for p near 0 or 1, revealing new monotonicity phenomena in anisotropic percolation models.

Findings

Connectivity function is monotone in n when p is close to 0.

Truncated connectivity function is monotone in n when p is close to 1.

Results apply to anisotropic bond percolation on hypercubic lattices.

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.