Scaling for the Percolation Backbone

TL;DR

This paper investigates the scaling behavior of the backbone connecting two sites in a 2D lattice percolation system, deriving new scaling laws and probability distributions for backbone mass.

Contribution

It introduces a scaling form for the average backbone mass and probability distribution, with new exponents and relations specific to the backbone structure in 2D percolation.

Findings

The average backbone mass scales as $L^{d_B- ext{psi}} r^{ ext{psi}}$.

The probability distribution $P(M_B)$ exhibits a power-law decay with exponent $ au_B \\approx 1.20$.

The exponents satisfy the relation $ ext{psi} = d_B ( au_B - 1)$.

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, $\psi=d-d_B$.