Universality of the crossing probability for the Potts model for q=1, 2, 3, 4
Abstract
The universality of the crossing probability π_{hs} of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the qstate Potts model for q=2, 3, 4 and for percolation q=1. We check the percolation through FortuinKasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated sitebond percolation model. It was shown that probability of a system to percolate only in the horizontal direction π_{hs} has the universal form π_{hs}=A(q)Q(z) for q=1,2,3,4 as a function of the scaling variable z={b(q)L^{1/ν(q)}[pp_{c}(q,L)]}^{ζ(q)}. Here, p=1exp(β) is the probability of a bond to be closed, A(q) is the nonuniversal crossing amplitude, b(q) is the nonuniversal metric factor, ν(q) is the correlation length index, and ζ(q) is the additional scaling index. The universal function Q(x)≃exp(z). The nonuniversal scaling factors were found numerically.
 Publication:

Physical Review E
 Pub Date:
 August 2003
 DOI:
 10.1103/PhysRevE.68.026125
 arXiv:
 arXiv:condmat/0204398
 Bibcode:
 2003PhRvE..68b6125V
 Keywords:

 64.60.Ak;
 05.10.Ln;
 05.70.Jk;
 Renormalizationgroup fractal and percolation studies of phase transitions;
 Monte Carlo methods;
 Critical point phenomena;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 15 pages, 3 figures, revtex4b, (minor errors in text fixed, journalref added)