The Brownian Bridge Asymptotics in the Subcritical Phase of Bernoulli Bond Percolation Model

#### Y. Kovchegov

2004, v.10, Issue 2, 327-344

ABSTRACT

For a given point $\vec{\math {a}}$ in $Z^d$, we prove that a cluster in the $d$-dimensional subcritical Bernoulli bond percolation model conditioned on connecting points $(0,\dots,0)$ and $n \vec{\math {a}}$ if scaled by $1 / (n \| \vec{\math {a}} \|)$ along $\vec{\math {a}}$ and by $1 / \sqrt{n}$ in the orthogonal directions converges asymptotically to Time $\times$ ($d-1$)-dimensional Brownian bridge.

Keywords: percolation,Brownian bridge,cluster