FINITE-SIZE EFFECTS IN ONE-DIMENSIONAL PERCOLATION - VERIFICATION OF SCALING THEORY

The critical behavior of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ xi , xi being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed,and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by theta equals lambda equals 1.