Scaling for the critical percolation backbone

We study the backbone connecting two given sites of a two-dimensional lattice separated by an arbitrary distance [Formula Presented] in a system of size [Formula Presented] at the percolation threshold. We find a scaling form for the average backbone mass: [Formula Presented] where [Formula Presented] can be well approximated by a power law for [Formula Presented] [Formula Presented] with [Formula Presented] This result implies that [Formula Presented] for the entire range [Formula Presented] We also propose a scaling form for the probability distribution [Formula Presented] of backbone mass for a given [Formula Presented] For [Formula Presented] [Formula Presented] is peaked around [Formula Presented] whereas for [Formula Presented] [Formula Presented] decreases as a power law, [Formula Presented] with [Formula Presented] The exponents ψ and [Formula Presented] satisfy the relation [Formula Presented] and ψ is the codimension of the backbone, [Formula Presented].