Scaling and finite-size effects for the critical backbone

In a first part, 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 and we also propose a scaling form for the probability distribution P(M_B) of backbone mass for a given r. For r ≈ L, P (M_B) is peaked around L^dB, whereas for r ≪ L, P (M_B) decreases as a power law, M_B^-τB, with τ_B≃1.20±0.03. The exponents ψ and τ_B satisfy the relation ψ = d_B(τ_B - 1) and ψ is the codimension of the backbone, ψ = d - d_B. In a second part, we study the multifractal spectrum of the current in the two-dimensional random resistor network at the percolation threshold. Our numerical results suggest that in the infinite system limit, the probability distribution behaves for small i as P(i) ∼ 1/i where i is the current. As a consequence, the moments of i of order q ≤ q_c = 0 diverge with system size, and all sets of bonds with current values below the most probable one have the fractal dimension of the backbone. Hence we hypothesize that the backbone can be described in terms of only (i) blobs of fractal dimension d_B and (ii) high current carrying bonds of fractal dimension going from d_red to d_B, where d_red is the fractal dimension of the red bonds carrying the maximal current.