The variance of the incipient infinite cluster in two-dimensional percolation

Abstract

Consider bond percolation on the square lattice. Let $\mathcal{C}$ be the incipient infinite cluster with the incipient measure $\nu$. If a one-arm path exponent exists and equals $5/48$, it is well known that $E_{\nu }|\mathcal{C}\cup [-n,n{]}^2|=n^{91/48+o(1)}$. In this paper, we focus on the variance of $|\mathcal{C}\cup [-n,n{]}^2|$ and show that ${\sigma }_{\nu }^2(|\mathcal{C}\cup [-n,n{]}^2|)=n^{91/24+o(1)}$.

14 pages and two figures

References in corpus (1)

• Path Crossing Exponents and the External Perimeter in 2D Percolation