Scaling for the critical percolation backbone.

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 at the percolation threshold. We find a scaling form for the average backbone mass: <M(B)> approximately L(dB)G(r/L), where G can be well approximated by a power law for 0<or=x<or=1: G(x) approximately x(psi) with psi=0.37+/-0.02. This result implies that <M(B)> approximately L(dB-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 approximately L, P(MB) is peaked around L(dB), whereas for r<<L, P(M(B)) decreases as a power law, M(-tauB)B, with tauB approximately 1.20+/-0.03. The exponents psi and tauB satisfy the relation psi=dB(tauB-1), and psi is the codimension of the backbone, psi=d-dB.