Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0005452

Scaling of crossing probabilities for the q-state Potts model at criticality

Oleg A. Vasilyev

Published May 25, 2000• 0 citations
arXivHTMLPDF

Abstract

We present study of finite-size scaling and universality of crossing probabilities for the $q$-state Potts model. Crossing probabilities of the Potts model are similar ones in percolation problem. We numerically investigated scaling of $π_{s}$ - 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 $q=3,4,5,6,8,10$. We found the thermal scaling index $y= \frac{1}ν$ for $q<4$. In contrast, $y \ne \frac{1}ν$ for $q=4$.

DOI
10.48550/arxiv.cond-mat/0005452

Related papers

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

Oleg A. Vasilyev

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.

arXiv:cond-mat/0204398•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 25, 2003•3 citations

Tails of the crossing probability

Oleg A. Vasilyev

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 ($q$-state Potts model) for $q=1$, 2, 3, 4 (where $q$ is the number of spin states). We have to demonstrate that the crossing probability ${\ensuremath{\pi}}_{hs}(p)$ far from the critical point ${p}_{c}$ 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 $x$ is a scaling index describing the central part of the crossing probability.

arXiv:cond-mat/0402294•Physical Review E•Sep 15, 2005•1 citation

Universality of the complete-graph Potts model with 0 < q ≤ 2

Zirui Peng, Sheng Fang, Hao Hu +1 more

Universality is a fundamental concept in modern physics. For the q-state Potts model, the critical exponents are merely determined by the order-parameter symmetry S_{q}, spatial dimensionality and interaction range, independent of microscopic details. In a simplest and mean-field treatment, i.e., the Potts model on complete graph (CG), the phase transition is further established to be of percolation universality for the range of 0<q<2. By simulating the CG Potts model in the random-cluster representation, we numerically demonstrate such a hyperuniversality that the critical exponents are the same for 0<q<2 and, moreover, the Ising system (q=2) exhibits a variety of critical geometric properties in percolation universality. On the other hand, many other universal properties in the finite-size scaling (FSS) theory, including Binder-like ratios and distribution function of the order parameter, are observed to be q dependent. Meanwhile, we have made improvements to the Monte Carlo algorithms for efficiently simulating the CG Potts model. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.

arXiv:2501.16930•Physical review. E•May 23, 2025•1 citation

Percolation, finite-size scaling, and the thermal scaling power for the Potts model

Chin‐Kun Hu, Chi-Ning Chen

Based on the connection between the q-state Potts model (QPM) and q-state bond-correlated percolation model, a simple geometrical method to calculate the thermal scaling power ${y}_{t}$ for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.

Physical review. B, Condensed matter•Jul 1, 1989•1 citation

Finite-size scaling of the free energy levels in the Ising and Potts q=3 universality

Loïc Turban, Jean‐Marc Debierre

The finite-size scaling of the free energy levels of Privman and Fisher (1984) is tested numerically on two-dimensional lattice strips in the Ising and q=3 Potts universality. The universal amplitudes and the free energy per site are obtained.

Journal of Physics A Mathematical and General•Apr 21, 1986•1 citation

Finite-size effects in one-dimensional percolation: a verification of scaling theory

Stanley Greenspoon

The critical behaviour 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/ξ, ξ 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 θ = λ = 1.

Canadian Journal of Physics•Apr 1, 1979•3 citations
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.
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=2,3,4q=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 ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for q=1,2,3,4q=1,2,3,4q=1,2,3,4 as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed, A(q)A(q)A(q) is the nonuniversal crossing amplitude, b(q)b(q)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.
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.
Universality is a fundamental concept in modern physics. For the q-state Potts model, the critical exponents are merely determined by the order-parameter symmetry S_{q}, spatial dimensionality and interaction range, independent of microscopic details. In a simplest and mean-field treatment, i.e., the Potts model on complete graph (CG), the phase transition is further established to be of percolation universality for the range of 0<q<2. By simulating the CG Potts model in the random-cluster representation, we numerically demonstrate such a hyperuniversality that the critical exponents are the same for 0<q<2 and, moreover, the Ising system (q=2) exhibits a variety of critical geometric properties in percolation universality. On the other hand, many other universal properties in the finite-size scaling (FSS) theory, including Binder-like ratios and distribution function of the order parameter, are observed to be q dependent. Meanwhile, we have made improvements to the Monte Carlo algorithms for efficiently simulating the CG Potts model. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.
Based on the connection between the q-state Potts model (QPM) and q-state bond-correlated percolation model, a simple geometrical method to calculate the thermal scaling power yt{y}_{t}yt​ for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.
The critical behaviour 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/ξ, ξ 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 θ = λ = 1.