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(MB) of backbone mass for a given r. For r ≈ L, P(MB) is peaked around LdB, whereas for r ≪ L, P(MB) decreases as a power law, [Formula: see text], with τB ≃ 1.20 ± 0.03. The exponents ψ and τB satisfy the relation ψ = dB(τB - 1), and ψ is the codimension of the backbone, ψ = d - dB. 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 ≤ qc = 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 dB and (ii) high current carrying bonds of fractal dimension going from d red to dB, where d red is the fractal dimension of the red bonds carrying the maximal current.
Gas-liquid phase coexistence and finite-size effects in a two-dimensional Lennard-Jones system
