A remark on monotonicity in Bernoulli bond Percolation

Abstract

Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\ge 2$, 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.

6 pages

References in corpus (1)

• Truncated Connectivities in a highly supercritical anisotropic percolation model

Cited by in corpus (1)

• Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach