Fractality of largest clusters and the percolation transition in power-law diluted chains

Relying on the fractal character of the largest clusters at criticality, we employ a finite-size scaling analysis to obtain an accurate phase-diagram of the percolation transition in chains with bond concentration decaying as a power-law on the form 1/r(1+sigma). For the particular case of sigma=1, no percolation transition is observed to occur at a finite dilution, in contrast with the finite temperature Kosterlitz-Thouless transition exhibited in Ising and Potts chains with inverse square-law couplings. The fractal dimension of the critical percolation cluster is found to follow distinct dependencies on the decay exponent being numerically fitted by d(f)=0.35+4 sigma/5 for 0 <sigma < 1/2 and d(f)=(1+sigma)/2 for 1/2 <sigma < 1. We also compute average mass ratios of the two largest clusters at criticality.