Ornstein-Zernike theory for the Bernoulli bond percolation on $\mathbb{Z}^d$

Abstract

We derive a precise Ornstein–Zernike asymptotic formula for the decay of the two-point function $\mathbb{P}_p (0 \leftrightarrow x)$ of the Bernoulli bond percolation on the integer lattice $\mathbb{Z}^d$ in any dimension $d \geq 2$, in any direction $x$ and for any subcritical value of $p < p_c (d)$.