Scaling for the critical percolation backbone

We study the backbone connecting two given sites of a two-dimensional latti ce separated by an arbitrary distance r in a system of size L at the percol ation threshold. We find a scaling form for the average backbone mass: [M-B ] similar to L(B)(d)G(r/L), where G can be well approximated by a power law for 0 less than or equal to r less than or equal to 1: G(x) similar to x(p si) with psi = 0.37 +/- 0.02. This result implies that [M-B] similar to L(B )(d)(-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 approximate to L, P(M-B) is peaked around L-B(d), whereas for r muc h less than L, P(M-B) decreases as a power law, M-B(-tau B), with tau(B) si milar or equal to 1.20 +/- 0.03. The exponents psi and tau(B) satisfy the r elation psi = d(B)(tau(B) - 1), and psi is the codimension of the backbone, psi = d - d(B). [S1063-651X(99)51408-6].