Critical behavior of the site percolation model on the square lattice in a L×M geometry

Abstract

Relevant aspects of the critical behavior of the site percolation model in a L×M geometry (L≪M) are studied. It is shown that this geometry favors the growth of percolating clusters in the L-direction with respect to those growing in the M-direction, causing pronounced finite-size effects on the percolation probabilities. Scaling functions have an additional parameter, namely M, which introduces a dependence of these functions on the aspect ratio L/M. At criticality, the probability of a site belonging to the percolation clusters (PL,M) behaves like PL,M∝Lβ/vφ(L/M) with β=5/36 and v=4/3, where φ is a suitable scaling function. Using scaling arguments it is conjectured and then tested by means of Monte Carlo simulations, the following asymptotic behavior φ(L/M)∝(L/M)δ, (L→∞,M→∞, δ=1), for the leading term. Systematic deviations of the Monte Carlo data from the conjectured behavior are due to second order corrections to the leading term which can also be under-stood on the basis of scaling ideas. Finite-size dependent “critical probabilities” are also functions of L/M as it follows from scaling arguments which are corroborated by the simulations.