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arXiv:cond-mat/0204398 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universality of the crossing probability for the Potts model for q=1,2,3,4

Oleg A. Vasilyev

Published Aug 25, 2003• 3 citations
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Abstract

The universality of the crossing probability ${\ensuremath{\pi}}_{\mathrm{hs}}$ of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for $q=2,3,4$ and for percolation $q=1.$ We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for $q=1,2,3,4$ as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed, $A(q)$ is the nonuniversal crossing amplitude, $b(q)$ is the nonuniversal metric factor, $\ensuremath{\nu}(q)$ is the correlation length index, and $\ensuremath{\zeta}(q)$ is the additional scaling index. The universal function $Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).$ The nonuniversal scaling factors were found numerically.

DOI
10.1103/physreve.68.026125

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The universality of the crossing probability \ensuremathπhs{\ensuremath{\pi}}_{\mathrm{hs}}\ensuremathπhs​ of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for q=2,3,4q=2,3,4q=2,3,4 and for percolation q=1.q=1.q=1. We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction \ensuremathπhs{\ensuremath{\pi}}_{\mathrm{hs}}\ensuremathπhs​ has the universal form \ensuremathπhs=A(q)Q(z){\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)\ensuremathπhs​=A(q)Q(z) for q=1,2,3,4q=1,2,3,4q=1,2,3,4 as a function of the scaling variable z=b(q)L1/\ensuremathν(q)[p\ensuremath−pc(q,L)]\ensuremathζ(q).z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.z=b(q)L1/\ensuremathν(q)[p\ensuremath−pc​(q,L)]\ensuremathζ(q). Here, p=1\ensuremath−exp(\ensuremath−\ensuremathβ)p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})p=1\ensuremath−exp(\ensuremath−\ensuremathβ) is the probability of a bond to be closed, A(q)A(q)A(q) is the nonuniversal crossing amplitude, b(q)b(q)b(q) is the nonuniversal metric factor, \ensuremathν(q)\ensuremath{\nu}(q)\ensuremathν(q) is the correlation length index, and \ensuremathζ(q)\ensuremath{\zeta}(q)\ensuremathζ(q) is the additional scaling index. The universal function Q(x)\ensuremath≃exp(\ensuremath−∣z∣).Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).Q(x)\ensuremath≃exp(\ensuremath−∣z∣). The nonuniversal scaling factors were found numerically.
The scaling of the tails of the probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{hs}$ was investigated numerically for the correlated site-bond percolation model (qqq-state Potts model) for q=1q=1q=1, 2, 3, 4 (where qqq is the number of spin states). We have to demonstrate that the crossing probability ${\ensuremath{\pi}}_{hs}(p)$ far from the critical point pc{p}_{c}pc​ has the shape ${\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}D\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[cL{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]$ where $\ensuremath{\nu}$ is the correlation length index, and $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed. For the tail region the correlation length is smaller than the lattice size. At criticality the correlation length reaches the sample size and we observe crossover to another scaling ${\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}{\ensuremath{-}b{[L{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]}^{x}}$. Here xxx is a scaling index describing the central part of the crossing probability.
Scale-invariant universal crossing probabilities are studied for critical anisotropic systems in two dimensions. For weakly anisotropic standard percolation in a rectangular-shaped system, Cardy's exact formula is generalized using a length-rescaling procedure. For strongly anisotropic systems in 1+1 dimensions, exact results are obtained for the random walk with absorbing boundary conditions, which can be considered as a linearized mean-field approximation for directed percolation. The bond and site directed percolation problem is itself studied numerically via Monte Carlo simulations on the diagonal square lattice with either free or periodic boundary conditions. A scale-invariant critical crossing probability is still obtained, which is a universal function of the effective aspect ratio r eff = c r , where r = L / t z , z is the dynamical exponent and c is a non-universal amplitude.
We present study of finite-size scaling and universality of crossing probabilities for the qqq-state Potts model. Crossing probabilities of the Potts model are similar ones in percolation problem. We numerically investigated scaling of πsπ_{s}πs​ - the probability of a system to percolate only in one direction for two-dimensional site percolation, the Ising model, and the q-state Potts model for q=3,4,5,6,8,10q=3,4,5,6,8,10q=3,4,5,6,8,10. We found the thermal scaling index y=1νy= \frac{1}νy=ν1​ for q<4q<4q<4. In contrast, y≠1νy \ne \frac{1}νy=ν1​ for q=4q=4q=4.
Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep,{E}_{p},Ep​, the percolation probability P,P,P, and the probability Wn{W}_{n}Wn​ for the appearance of nnn percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L.L.L. Using the HMCSM, we calculate the existence probability Ep{E}_{p}Ep​ and the percolation probability PPP for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep{E}_{p}Ep​ and PPP in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep{E}_{p}Ep​ and PPP have universal finite-size scaling functions. This implies that the critical Ep{E}_{p}Ep​ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that Wn{W}_{n}Wn​ for site and bond percolation on sc lattices have universal finite-size scaling functions.
Reexamining algebraic curves found in the eight-vertex model, we propose an asymptotic form of the correlation functions for off-critical systems possessing rotational and mirror symmetries of the square lattice, i.e., the C_{4v} symmetry. In comparison with the use of the Ornstein-Zernike form, it is efficient to investigate the correlation length with its directional dependence (or anisotropy). We investigate the Q-state Potts model on the square lattice. Monte Carlo (MC) simulations are performed using the infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with the asymptotic form above the critical temperature, we reproduce the exact solution of the the anisotropic correlation length (ACL) of the Ising model (Q=2) within a five-digit accuracy. For Q=3 and 4, we obtain numerical evidence that the asymptotic form is applicable to their correlation functions and the ACLs. Furthermore, we successfully apply it to the bond percolation problem which corresponds to the Q→1 limit. From the calculated ACLs, the equilibrium crystal shapes (ECSs) are derived via duality and Wulff's construction. Regarding Q as a continuous variable, we find that the ECS of the Q-state Potts model is essentially the same as those of the Ising models on the Union Jack and 4-8 lattices, which are represented in terms of a simple algebraic curve of genus 1.
We present a method which allows calculation of whole universal finite-size-scaling functions from Monte Carlo data. The crux of the new method is a technique of isolating the singular part from the total free energy. We apply this method to the 3-state Potts model on a square lattice and find the normalized scaling function Y(x) in the form 1+x+5.31x2{\mathit{x}}^{2}x2-1.2x3{\mathit{x}}^{3}x3-0.67x4{\mathit{x}}^{4}x4 . . . near the bulk critical point x=0, together with the normalizing universal amplitude E=1.053 and a nonuniversal metric factor D=-0.387. It also behaves as Y(x)\ensuremath{\approxeq}(\ifmmode\pm\else\textpm\fi{}x${)}^{\mathit{d}\ensuremath{\nu}}$${\mathit{Q}}_{\ifmmode\pm\else\textpm\fi{}}$ for large x in agreement with the hyperuniversality hypothesis with Q+{\mathit{Q}}_{+}Q+​/${\mathit{Q}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\approxeq}1.