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/9709285 · Physical review. B, Condensed matter

Exact results at the two-dimensional percolation point

Peter H. Kleban, Robert M. Ziff

Published Apr 1, 1998• 36 citations
arXivHTMLPDF

Abstract

We derive exact expressions for the excess number of clusters $b$ and the excess cumulants ${b}_{n}$ of a related quantity at the two-dimensional percolation point. High-accuracy computer simulations are in accord with our predictions. The parameter $b$ is a finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula for the number of clusters per site ${n}_{c}$ in the infinite system limit; the ${b}_{n}$ correct bulk cumulants. Both $b$ and ${b}_{n}$ are universal, and thus depend only on the system's shape. Higher-order corrections show no apparent dependence on fractional powers of the system size.

DOI
10.1103/physrevb.57.r8075

Related papers

Universality of Finite-Size Corrections to the Number of Critical Percolation Clusters

Robert M. Ziff, Steven R. Finch, Victor Adamchik

Monte Carlo simulations on a variety of finite 2D percolating systems at criticality suggest that the excess number of clusters over the bulk value ${n}_{c}$ is a universal quantity, dependent upon the system shape but independent of the lattice and percolation type. Values of ${n}_{c}$ are found to high accuracy, and for bond percolation are in accord with the theoretical predictions of Temperley and Lieb [Proc. R. Soc. London A 322, 251 (1971)], and Baxter, Temperley, and Ashley [Proc. R. Soc. London A 358, 535 (1978)], whose results we have evaluated explicitly in terms of simple algebraic numbers. Fluctuations are also studied.

arXiv:cond-mat/9707168•Physical Review Letters•Nov 3, 1997•73 citations

Scaling studies of percolation phenomena in systems of dimensionality two to seven: Cluster numbers

Hisao Nakanishi, Eugene Stanley

Cluster statistics are obtained by computer simulation for percolation processes on $d$-dimensional lattices with $d=2$ through 7. For all $d$, ${n}_{s}$, the number of $s$-site clusters per site, is found to satisfy reasonably well the scaling hypothesis first proposed by Stauffer. The scaling functions are analyzed for dimensional dependence, and it is found that as $d$ increases they approach very rapidly the exactly known result for the Bethe lattice corresponding to $d=\ensuremath{\infty}$. Corrections to scaling are also studied, and at the upper critical dimension ${d}_{c}=6$, a deviation from scaling consistent with a logarithmic correction is obtained. Some universal quantities such as the ratio of the amplitudes ${C}_{+}$ and ${C}_{\ensuremath{-}}$ of the "susceptibility" (second moment of ${n}_{s}$) below and above the percolation threshold ${p}_{c}$ are also found to approach the Bethe lattice limit very quickly. In addition, our data suggest that $\frac{{C}_{+}}{{C}_{\ensuremath{-}}}$ already assumes the limiting value of unity for $d=6$. This is consistent with the exact relation for the asymmetric decay of ${n}_{s}$ proved by Kunz and Souillard if we adopt the hypothesis that the asymmetry of ${n}_{s}$ about ${p}_{c}$ only enters the corrections to the leading scaling term for $d>~6$.

Physical review. B, Condensed matter•Sep 1, 1980•110 citations

Bond and site percolation in three dimensions

Junfeng Wang, Zongzheng Zhou, Wei Zhang +2 more

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to $L=512$, and estimate the percolation thresholds to be ${p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)$ and ${p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)$. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent ${d}_{\mathrm{min}}=1.3756(3)$. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

arXiv:1302.0421•Physical Review E•May 7, 2013•170 citations

Test of universal finite-size scaling in two-dimensional site percolation

Amnon Aharony A, Dietrich Stauffer

Traditional two-scale-factor universality, concerning the number of clusters within a correlation volume near the percolation threshold, is reconfirmed by a comparison of site percolation on square and triangular lattices. We also test universality at the threshold, e.g. of the ratio , where is the correlation length and L the lattice size. At the threshold, the result is sensitive to the system's shape and aspect ratio, boundary conditions, the algorithm, the detailed units measuring and L, and possibly other factors as yet unexplored.

Journal of Physics A Mathematical and General•May 21, 1997•20 citations

Universality of percolation at dynamic pseudocritical point

Qiyuan 骐源 Shi 石, Su‐Huai Wei, Youjin Deng +1 more

Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.

arXiv:2502.01121•Chinese Physics B•Jan 22, 2025•3 citations

Percolation in the canonical ensemble

Hao Hu, Henk W. J. Blöte, Youjin Deng

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y can = 2 y t − d both in energy-like and magnetic quantities, where y t = 1/ν is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of the universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y can = −1/2. This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

arXiv:1210.3463•Journal of Physics A Mathematical and Theoretical•Nov 27, 2012•20 citations
We derive exact expressions for the excess number of clusters bbb and the excess cumulants bn{b}_{n}bn​ of a related quantity at the two-dimensional percolation point. High-accuracy computer simulations are in accord with our predictions. The parameter bbb is a finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula for the number of clusters per site nc{n}_{c}nc​ in the infinite system limit; the bn{b}_{n}bn​ correct bulk cumulants. Both bbb and bn{b}_{n}bn​ are universal, and thus depend only on the system's shape. Higher-order corrections show no apparent dependence on fractional powers of the system size.
Monte Carlo simulations on a variety of finite 2D percolating systems at criticality suggest that the excess number of clusters over the bulk value nc{n}_{c}nc​ is a universal quantity, dependent upon the system shape but independent of the lattice and percolation type. Values of nc{n}_{c}nc​ are found to high accuracy, and for bond percolation are in accord with the theoretical predictions of Temperley and Lieb [Proc. R. Soc. London A 322, 251 (1971)], and Baxter, Temperley, and Ashley [Proc. R. Soc. London A 358, 535 (1978)], whose results we have evaluated explicitly in terms of simple algebraic numbers. Fluctuations are also studied.
Cluster statistics are obtained by computer simulation for percolation processes on ddd-dimensional lattices with d=2d=2d=2 through 7. For all ddd, ns{n}_{s}ns​, the number of sss-site clusters per site, is found to satisfy reasonably well the scaling hypothesis first proposed by Stauffer. The scaling functions are analyzed for dimensional dependence, and it is found that as ddd increases they approach very rapidly the exactly known result for the Bethe lattice corresponding to $d=\ensuremath{\infty}$. Corrections to scaling are also studied, and at the upper critical dimension dc=6{d}_{c}=6dc​=6, a deviation from scaling consistent with a logarithmic correction is obtained. Some universal quantities such as the ratio of the amplitudes C+{C}_{+}C+​ and ${C}_{\ensuremath{-}}$ of the "susceptibility" (second moment of ns{n}_{s}ns​) below and above the percolation threshold pc{p}_{c}pc​ are also found to approach the Bethe lattice limit very quickly. In addition, our data suggest that $\frac{{C}_{+}}{{C}_{\ensuremath{-}}}$ already assumes the limiting value of unity for d=6d=6d=6. This is consistent with the exact relation for the asymmetric decay of ns{n}_{s}ns​ proved by Kunz and Souillard if we adopt the hypothesis that the asymmetry of ns{n}_{s}ns​ about pc{p}_{c}pc​ only enters the corrections to the leading scaling term for d> 6d>~6d> 6.
We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512L=512L=512, and estimate the percolation thresholds to be pc(bond)=0.24881182(10){p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)pc​(bond)=0.24881182(10) and pc(site)=0.3116077(2){p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)pc​(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent dmin=1.3756(3){d}_{\mathrm{min}}=1.3756(3)dmin​=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
Traditional two-scale-factor universality, concerning the number of clusters within a correlation volume near the percolation threshold, is reconfirmed by a comparison of site percolation on square and triangular lattices. We also test universality at the threshold, e.g. of the ratio , where is the correlation length and L the lattice size. At the threshold, the result is sensitive to the system's shape and aspect ratio, boundary conditions, the algorithm, the detailed units measuring and L, and possibly other factors as yet unexplored.
Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.
We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y can = 2 y t − d both in energy-like and magnetic quantities, where y t = 1/ν is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of the universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y can = −1/2. This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.