Universality of percolation at dynamic pseudocritical point
Qiyuan Shi(石骐源)1, Shuo Wei(魏硕)1, Youjin Deng(邓友金)2,1,3,†, and Ming Li(李明)4,‡
1 Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China; 2 Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China; 3 Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China; 4 School of Physics, Hefei University of Technology, Hefei 230009, ChinaResearch Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China; 3 Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China; 4 School of Physics, Hefei University of Technology, Hefei 230009, China
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.
Fund: The authors acknowledge helpful discussions with Jingfang Fan. The research was supported by the National Natural Science Foundation of China (Grant No. 12275263), the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0301900), and Natural Science Foundation of Fujian Province of China (Grant No. 2023J02032).
Corresponding Authors:
Youjin Deng
E-mail: yjdeng@ustc.edu.cn;lim@hfut.edu.cn
Cite this article:
Qiyuan Shi(石骐源), Shuo Wei(魏硕), Youjin Deng(邓友金), and Ming Li(李明) Universality of percolation at dynamic pseudocritical point 2025 Chin. Phys. B 34 040501
[1] Stauffer D and Aharony A 1991 Introduction to percolation theory (Taylor & Francis) [2] Kim S, Choi S, Oh E, Byun J, Kim H, Lee B, Lee S and Hong Y 2016 Sci. Rep. 6 34632 [3] Li M, Liu R R, Lü L, Hu M B, Xu S and Zhang Y C 2021 Phys. Rep. 907 1 [4] Ziff R M 2021 Physica A 568 125723 [5] Sykes M F and Essam J W 1964 J. Math. Phys. 5 1117 [6] Lieb E H 1967 Phys. Rev. Lett. 18 692 [7] Baxter R J 1972 Ann. Phys. 70 193 [8] Belavin A, Polyakov A and Zamolodchikov A 1984 Nucl. Phys. B 241 333 [9] Friedan D, Qiu Z and Shenker S 1984 Phys. Rev. Lett. 52 1575 [10] Nienhuis B 1984 J. Stat. Phys. 34 731 [11] Cardy J L 1986 Physica A 140 219 [12] Cardy J 2005 Ann. Phys. 318 81 [13] Smirnov S and Werner W 2001 Math. Res. Lett. 8 729 [14] Grassberger P, Christensen C, Bizhani G, Son S W and Paczuski M 2011 Phys. Rev. Lett. 106 225701 [15] D’Souza R M and Nagler J 2015 Nat. Phys. 11 531 [16] Achlioptas D, D’Souza R M and Spencer J 2009 Science 323 1453 [17] Friedman E J and Landsberg A S 2009 Phys. Rev. Lett. 103 255701 [18] Ziff R M 2009 Phys. Rev. Lett. 103 045701 [19] D’Souza R M and Mitzenmacher M 2010 Phys. Rev. Lett. 104 195702 [20] Li M, Wang J and Deng Y 2023 Phys. Rev. Lett. 130 147101 [21] Li M, Wang J and Deng Y 2024 Phys. Rev. Research 6 033319 [22] Lu M, Fang S, Zhou Z and Deng Y 2024 Phys. Rev. E 110 044140 [23] Kenna R and Berche B 2017 J. Phys. A: Math. Theor. 50 235001 [24] Borgs C, Chayes J T, van der Hofstad R, Slade G and Spencer J 2005 Random Struct. Algorithms 27 137 [25] Borgs C, Chayes J T, Slade G, Spencer J and van der Hofstad R 2005 Ann. Probab. 33 1886 [26] Heydenreich M and Van Der Hofstad R 2007 Commun. Math. Phys. 270 335 [27] Aizenman M 1997 Nucl. Phys. B 485 551 [28] Li M, Fang S, Fan J and Deng Y 2024 arXiv:2412.06228 [29] Fan J, Meng J, Liu Y, Saberi A A, Kurths J and Nagler J 2020 Nat. Phys. 16 455 [30] Langlands R P, Pichet C, Pouliot P and Saint-Aubin Y 1992 J. Stat. Phys. 67 553 [31] Pinson H T 1994 J. Stat. Phys. 75 1167 [32] Ziff R M, Lorenz C D and Kleban P 1999 Physica A 266 17 [33] Binder K 1981 Phys. Rev. Lett. 47 693 [34] Binder K 1981 Z. Phys. B 43 119 [35] Ouyang Y, Deng Y and Blöte H W J 2018 Phys. Rev. E 98 062101 [36] Newman M E J and Ziff R M 2000 Phys. Rev. Lett. 85 4104 [37] Mendelson K S 1999 Phys. Rev. E 60 6496 [38] Deng Y, Ouyang Y and Blöte H W J 2019 J. Phys.: Conf. Ser. 1163 012001 [39] Feshanjerdi M and Grassberger P 2024 arXiv:2401.05234 [40] Hu H, Blöte HWand Deng Y 2012 J. Phys. A: Math. Theor. 45 494006 [41] Qian X, Deng Y and Blöte H W J 2005 Phys. Rev. B 71 144303 [42] Deng Y and Blöte H W J 2005 Phys. Rev. E 71 016117 [43] Lu M, Song Y F, Li M and Deng Y 2024 arXiv:2411.04748 [44] Beirlant J, Goegebeur Y, Segers J and Teugels J L 2006 Statistics of extremes: Theory and applications (JohnWiley & Sons)
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
