Scaling for the Percolation Backbone

Abstract

We study the backbone connecting two given sites of a two-dimensional lattice separated by an arbitrary distance $r$ in a system of size $L$. We find a scaling form for the average backbone mass: $<M_B>\sim L^{d_B}G(r/L)$, where $G$ can be well approximated by a power law for $0\le x\le 1$: $G(x)\sim x^{\psi }$ with $\psi =0.37\pm 0.02$. This result implies that $<M_B>\sim L^{d_B-\psi }{r}^{\psi }$ for the entire range $0<r<L$. We also propose a scaling form for the probability distribution $P(M_B)$ of backbone mass for a given $r$. For $r\approx L,P(M_B)$ is peaked around $L^{d_B}$, whereas for $r\ll L,P(M_B)$ decreases as a power law, $M_B^{-{\tau }_B}$, with ${\tau }_B\simeq 1.20\pm 0.03$. The exponents $\psi$ and ${\tau }_B$ satisfy the relation $\psi =d_B({\tau }_B-1)$, and $\psi$ is the codimension of the backbone, $\psi =d-d_B$.

3 pages, 5 postscript figures, Latex/Revtex/multicols/epsf