Abstract
In this paper, a 40-year-old theorem is tested, that the incipient percolation cluster has a fractal dimension equal to 91/48. With the Newman–Ziff algorithm, we measure the mass M of the incipient percolation cluster (i.e., the size of the largest cluster at the percolation threshold) versus the linear system size L which (after averaging over 
system realizations) nicely follows the power law 
with exponents 
ranging from 1.893954 to 1.89823 for the square lattice. The obtained fractal dimension agrees well with its analytical partner and those confirmed numerically earlier for compact neighborhoods with the nearest-neighbors on triangular and square lattices and holds for other considered neighborhoods on square lattice, including those that are not-compact. With six digits of the accuracy of reaching the percolation threshold, the percentage error of the numerical values obtained for the fractal dimension ranges from 0.028‰ to 1.264‰, which strengthens the earlier results confirming the universality of the fractality of the incipient percolation cluster. Using the Hoshen–Kopelman algorithm for cluster identification for 
and the box-counting procedure for the evaluation of the fractal dimension, after 
system realizations, we reached the percentage error of the numerical values obtained for the fractal dimension from 5‰ to 7‰, which is much worse than the percentage error obtained directly from the mass of the incipient percolation cluster as a function of the linear size of the system. Our results indicate that universality of fractality of the incipient percolation cluster is valid also for complex (non-compact) neighborhoods, which allow for occupied site connections with more ‘holes’ in cluster than allowed for extended (compact) neighborhoods. Also for a simple cubic lattice we get 
—independently on assumed neighborhoods—although these values are slightly higher than known in literature.
Keywords: Fractal dimension, Random site percolation, Newman–Ziff algorithm, Hoshen–Kopelman algorithm, Complex and extended neighborhoods, Monte Carlo simulation
Subject terms: Physics; Condensed-matter physics; Statistical physics, thermodynamics and nonlinear dynamics
Introduction
One of the core problems in statistical physics is considered a percolation1–3, which is a purely geometrical example of a system that exhibits the phase transition. This phase transition (in the case of the so-called site percolation) occurs at the critical occupancy 
of the sites in the system. Below 
the system behaves as an insulator (does not allow any medium flow among the system borders—to keep the original nomenclature taken from the rheology, where the percolation phenomenon was originally described4), while for the probability of site occupancy p above 
the path connects the system borders (allowing liquid or current flow among them).
At 
the system of occupied sites—incipient percolation cluster (IPC)5—exhibits fractal properties (see Reference 6 for the most recent review), which means that its mass M scales with the linear size L of the system as
 |
1 |
instead of according to 
, where 
and d are the fractal dimension and the dimension of the space in which the system is embedded, respectively.
The theoretical value of the fractal dimension 
7–9 of the IPC for two-dimensional (
) lattices has universal character and it is equal [2, Equations (54b) and (49)], [10, Table I and Table III] to
 |
2 |
where 
, 
and 
are universal exponents responsible for the scaling properties of the system [2, p. 54]. Namely, the exponent 
[2, p. 31] scales the percolation strength, 
[2, Equation (25)] characterizes the exponential decay of the number of clusters of size s with this size, while the exponent 
[2, Equation (47c)] governs the scaling of the length of connectivity as p approaches 
. The fractal dimension of the IPC in two dimensions is equal to 
[2, p. 54], [6, p. 2], [8, p. 393]. As pointed out by Li and Deng11: “the fractal dimension 
in 2D is predicted by Coulomb gas arguments12,13, conformal field theory14,15, and stochastic Löwner evolution theory16,17, and it has been rigorously proven on the triangular lattice18”. The Coulomb gas formulation yield 
also earlier in Reference19. This value was also calculated in extensive numerical simulations first for the triangular lattice [2, p. 70] and then confirmed for the square lattice20.
The old concept of extended neighborhoods introduced by Dalton et al.21 and Domb and Dalton22 (and extensively studied by d’Iribarne et al.23) was recently revived in a series of articles dealing with the percolation process in various topologies and various dimensionalities of the system with compact24–29 and non-compact30–41 neighborhoods containing sites up to the ninth coordination zone (for compact neighborhoods) and up to the seventh coordination zone (for non-compact neighborhoods). The non-compact neighborhoods contain extra ‘holes’. The presence of these ‘holes’ can, in principle, change the fractality of the system (for example, to decrease the fractal dimension 
of the IPC).
In this paper, we check if universality of the fractal dimension 
of the IPC also holds for non-compact neighborhoods, that is, for those which contain above-mentioned ‘holes’. With computer simulations, we check if the fractal dimension 
of the IPC for square and simple cubic lattices and non-compact neighborhoods is still constant independently of complexity of neighborhood (for assumed lattice dimensionality). Our results show that in fact the universality of the fractal dimension holds also for complex (and also non-compact) neighborhoods.
The paper is organized as follows. In the next Section “Methods” we provide the methodology of our studies, then we present the results of computer simulations (in Section “Results”). The results obtained are summarized in Section “Conclusions”. Examples of IPC and examples of site labeling for various neighborhoods for square lattices are presented in the Supplemental Material42.
Methods
We calculate the fractal dimension 
of the IPC (i.e., the largest cluster of occupied sites for occupation probability 
) for various neighborhoods presented in Fig. 1 on square [Fig. 1(a)–(h)] and simple cubic [Fig. 1(i)–(ff)] lattices. For neighborhoods names, we keep the convention proposed in the Reference44. For a simple cubic lattice, we decided to use the partial neighborhoods sc-8* and sc-8† of sc-8 to be able to combine them with other neighborhoods to obtain the three-dimensional equivalence of sq-1,3,6 [Fig. 1(e)] and sq-1,3,4,5 [Fig. 1(d)], respectively. In this way, the neighborhood sc-1,4,8† [Fig. 1(q)–(t)] is as 3d equivalence of sq-1,3,6 (six sticks of length three) and the neighborhood sc-1,4,5,6,7,8*,11 [Fig. 1(x)–(z)] is 3d equivalence of sq-1,3,4,5 (as the faces of cubic accompanied with six sticks but with empty sites inside cubic).
Fig. 1.
The black dots indicate the sites in the neighborhood of the red one. Basic neighborhood (a) sq-1, 
43, extended neighborhood (b) sq-1,2, 
2,43, and four examples of complex neighborhoods: (c) sq-3,4,5, 
, (d) sq-1,3,4,5, 
, (e) sq-1,3,6, 
39,40, (f) sq-4,6, 
39,40 (g) sq-2,4,7, 
41, (h) sq-5,6,7, 
41. Cuttings for planes: 
; 
(and if necessary for 
; 
) in three-dimensional space constituting neighborhoods for simple cubic lattice with complex neighborhoods: (i)–(j) sc-1; (k)–(m) sc-7,
,11; (n)–(p) sc-4,5,6,7,
,11; (q)–(t) sc-1,4,
; (u)–(w) sc-1,4,7,
,11; (x)–(z) sc-1,4,5,6,7,
,11; (aa)–(bb) sc-1,2,3; (cc)–(ff) sc-3,6,
,10,13,17, where sc-
and sc-
are parts of sc-8 neighborhood at the site (x, y, z) but without some neighbors, namely sc-
sc-8
and sc-
sc-8
.
To calculate the fractal dimension 
, we measure the dependence of the mass M of the IPC as dependent on the linear lattice size L. Then, the least squares method was used to fit a straight line to the dependence 
according to Eq. (1).
The Newman–Ziff algorithm
To create IPC, knowledge of the percolation threshold 
is required. For sq-1 and sq-1,2 the values 
are known with a precision of 15 digits. Six-digit precision has been achieved for the sq-1,3,6, sq-4,6, sq-2,4,7 and sq-5,6,7 neighborhoods. The values of 
and 
are known only with 
accuracy, and we have to recalculate them to reach the accuracy as for the above-mentioned neighborhoods. Also, for a simple cubic lattice and various neighborhoods, we calculate percolation thresholds 
. To that end, we use the efficient Newman and Ziff Monte Carlo algorithm45 and the finite-size scaling theory46,47.
According to the finite-size scaling theory46,47 in the vicinity of the percolation threshold 
the probability of belonging to the largest cluster
 |
3 |
scales with the linear size of the system L and the distance to the critical point 
as
 |
4 |
where 
is the size (mass) of the largest cluster, the critical exponents 
, 
(for two-dimensional lattices) [2, p. 52], 
48, 
49 (for three-dimensional lattices), and 
is a scaling function. Please note that at 
the rescaled probability of belonging to the largest cluster 
is independent of the size of the system L. This allows us to estimate the percolation threshold 
by searching the common point of 
versus p plotted for various values of L.
The size 
of the largest cluster (presented, for example, in Fig. 2(b)) is calculated with the Newman and Ziff algorithm45. Fortunately, to get the percolation threshold, we first need 
, which for 
is exactly the required mass M of the IPC.
Fig. 2.
(a) Example of cluster labelling for sq-1,2, 
sites, 
. Various colors correspond to various clusters. (b) The largest cluster extracted from clusters given in the left panel. More examples of the shapes of the IPC for sq-1, sq-1,2, sq-1,3,6, sq-4,6, sq-3,4,5 and sq-1,3,4,5 are presented in black-and-white panels of Figures 1 to 6 in the Supplemental Material42.
The algorithm is fast and efficient as it is based on the recursive construction of the system with n occupied sites with the addition of only one occupied site to the system containing 
already occupied sites. Then convolution with the binomial (Bernoulli) probability distribution allows us to transform 
in the domain of the number n of occupied sites into 
in the domain p of the probability of site occupation. The immanent part of the Newman and Ziff algorithm45 is also a concept of the efficient construction of binomial coefficients.
The Hoshen–Kopelman algorithm
To get an idea of the shape of the IPC for a square lattice and various neighborhoods, we employ the Hoshen and Kopelman algorithm35,50,51. With the Hoshen and Kopelman algorithm, every site is labeled. Sites in the same cluster have the same labels, while the labels for various clusters are different. The main idea of this algorithm is to check every occupied site and its occupied neighbors. The analyzed site is labeled with lower label among occupied and so far labeled neighbors. In case when simultaneously two labels should be assigned, one should choose the lower one and remember that higher of them should be relabeled to lower one.
In Fig. 2(a) example of site labels for sq-1,2 and 
is presented. Various colors correspond to various labels. More examples of cluster labeling for the square lattice with neighborhoods sq-1, sq-1,2, sq-1,3,6, sq-4,6, sq-3,4,5 and sq-1,3,4,5 are presented in color panels of Figures 1 to 6 in the Supplementary Materials42, respectively. We note that similar pictures of the largest percolation cluster—but for the bond percolation problem—were presented in Figures 1–4 in Reference52.
The box-counting technique
Alternatively to Eq. (1), for the estimation of the fractal dimension 
one may apply the box-counting technique [8, pp. 240–243]. With this technique, to estimate the fractal dimension 
of a given set (figure in two-dimensional space), one needs to count the number B of all boxes that intersect (or even touch) the considered figure. This counting should be repeated for different side lengths b of the boxes. Then, the fractal dimension is recovered from
 |
5 |
dependence. The example of box-counting for the IPC (for sq-1,3,4,5 neighborhood, Figure 6(b) in the Supplemental Material42) is presented in Fig. 3.
Fig. 3.
Example of box counting technique. The black figure (the IPC for sq-1,3,4,5 neighborhood, Figure 6(b) in the Supplemental Material42) has one, three, ten, 35, 124 boxes that intersect or touch the figure for box side length 
, 4096, 2048, 1024 and 512, respectively.
Results
For the sq-3,4,5 and sq-1,3,4,5 neighborhoods, the precision of the estimation of 
is of the order 
, namely 
32,40 and 
32,40. Thus, with the technique described in Section “Methods” we improved their accuracy to the order 
. The thresholds 
obtained by inspection of the rescaled probability of belonging to the largest cluster 
as dependent on the site occupation probability p are 0.17288 [see Fig. 4(a)] and 0.17922 [see Fig. 4(b)], for sq-1,3,4,5 and sq-3,4,5, respectively.
Fig. 4.
Examples of rescaled probabilities 
of belonging to the largest cluster vs. occupation probability p for (a) sq-1,3,4,5, (b) sq-3,4,5, (c) sc-1,4,7,
,11, (d) sc-1,4,5,6,7,
,11 neighborhoods. The common crossing point indicates the 
. The estimated percolation thresholds are 
, 
, 
and 
.
For a simple cubic lattice, and the neighborhoods considered here, we are warned of the percolation threshold only for the nearest neighbors (with the most recent estimate of 
53) and sc-1,2,3 (with estimation of 
21, 
33 and 
24).
Using the technique described in Section “Methods” we estimate 
values with uncertainty of 
for the neighborhoods presented in Fig. 1(i)–(ff). The examples of the rescaled probability of belonging to the largest cluster 
as dependent on the site occupation probability p for simple cubic lattice are presented in Fig. 4(c)–(d). The obtained percolation thresholds are presented in Table 1.
Table 1.
Percolation threshold 
, fractal dimension 
of the IPC and its percentage error 
for various neighborhoods on square and simple cubic lattice
| Neighborhood |  |
 |
 |
|---|---|---|---|
| tr-1 | 1/22 | 1.92 |  |
| sq-1 | 0.5927443 | 1.893954(90) | 0.0991% |
| sq-1,2 | 0.407252,43 | 1.895740(52) | 0.0049% |
| sq-1,3,6 | 0.2257739,40 | 1.89611(14) |  |
| sq-4,6 | 0.1979939,40 | 1.89437(11) | 0.0772% |
| sq-3,4,5 | 0.17922 | 1.89578(13) | 0.0028% |
| sq-1,3,4,5 | 0.17288 | 1.89577(43) | 0.0033% |
| sq-2,4,7 | 0.1486841 | 1.89617(41) |  |
| sq-5,6,7 | 0.1484841 | 1.89823(30) |  |
| sc-1 | 0.3116076853 | 2.52288(66) | |
sc-1,2,3 (up to  ) |
0.097644424 | 2.5227(12) | |
sc-1,4, (up to  ) |
0.09545 | 2.5358(18) | |
sc-7, ,11
|
0.03611 | 2.5388(36) | |
sc-1,4,7, ,11
|
0.03322 | 2.53140(22) | |
sc-4,5,6,7, ,11
|
0.02233 | 2.5414(20) | |
sc-1,4,5,6,7, ,11
|
0.02219 | 2.53974(98) | |
sc-3,6, ,10,13,17 (up to  ) |
0.01300 | 2.5270(10) |
The first row indicates data also for triangular lattice with the nearest-neighbors interaction tr-1. Reference 20 reports 
for 
what results in 
.
The dependencies of 
for various L and for various neighborhoods are presented in Fig. 5(a) (for the square lattice) and Fig. 5(b) (for the simple cubic lattice). The values of 
according to the Newman and Ziff algorithm are averaged over 
simulations. The solid lines are the results of power-law fits (1) with the least squares method. The uncertainties in the estimation of these exponents represent the uncertainties of 
, that is, the uncertainties of the estimation of slope of the linear fit to the 
vs. 
dependency. The fractal dimensions obtained with their uncertainties are presented in Table 1.
Fig. 5.
Dependence of mass M of ICP on linear system size L at 
for (a) square and (b) simple cubic lattices. The least squares fit yields (a) 
, 
, 
, 
, 
, 
, 
, 
, (b) 
, 
, 
, 
, 
, 
, 
, 
.
The last column of Table 1 shows the percentage error54, p. 14
 |
6 |
of fractal dimension 
measured for the square lattice with respect to its theoretical partner 
.
Conclusions
The fractal dimension of the IPC in two dimensions has universal character (i.e., it does not depend on the details regarding the lattice or neighborhoods and depends only on the dimensionality of the system) and for 
it is equal to 91/48. This theoretical value was confirmed in an extensive numerical simulation first for the triangular [2, p. 70] lattice, then for the square20 lattice. In this paper, the fractal dimension 
of the IPC is also calculated for square and simple cubic lattices with various extended and complex neighborhoods.
Complex neighborhoods like sq-3,4,5 (Fig. 1(c)), sq-1,3,4,5 (Fig. 1(d)), sq-4,6 (Fig. 1(f)), sq-2,4,7 (Fig. 1(g)) and sq-5,6,7 (Fig. 1(h)) are non-compact, they contain ‘holes’ and thus in principle, potentially, can change (reduce) the fractal dimension 
of IPC below theoretically predicted value 
.
Our results show that even for non-compact neighborhoods, the percentage error (6) of the measured fractal dimension 
with respect to its theoretical partner 
does not exceed 0.1264%, which strengthens the hypothesis of universality of the fractal dimension for IPC. Also for the three-dimensional case we see very similar behavior: independently on assumed neighborhood the obtained values of the fractal dimension 
do not depend on assumed shape of neighborhood, even when it is both complex and non-compact. For the simple cubic lattice 
—independently on assumed neighborhoods—although these values are slightly higher than known in the literature for the simple cubic lattice with the nearest-neighbors interactions (
55, 2.530(4)56, 2.5230(1)57, 2.5226(1)58, 2.52293(10)53 and 2.52288(66) here). This spread of values 
comes from both: a not exactly known value of 
(which affects the values 
) and from the accuracy (
) of estimating values of 
.
Furthermore, for the square lattice, we used the Hoshen and Kopelman algorithm to generate lattices with sites occupied with probability 
as long as necessary to obtain 
lattices where the largest cluster spans from top to bottom of the lattice and the linear size of the lattice is 
. These 
IPC created for 
are taken into account for the averaging procedure to estimate the number B of boxes of side length b. The plots (not shown) of B(b) on a logarithmic scale allow the fitting of a straight line with its slope giving the fractal dimension 
. Independently of the considered neighborhood, the power law (5) holds well for three and a half decades, that is, for the box length 
. However, we note that with this technique the percentage error 
of the obtained values 
ranges from 5‰ to 7‰, i.e., is much higher than this given by direct fit 
from Eq. (1).
Both, for square and simple cubic lattices, the values of fractal dimension obtained here are very close to each other, strengthening theories that the fractal dimension of IPC depends only on the lattice dimension d, and not on the details regarding the lattice, including the assumed neighborhood.
Further studies may concentrate on confirmation of the fractal dimension value 
of the IPC for non-compact neighborhoods for other two-dimensional lattices, like triangular36,37,40 or honeycomb38,40, where the percolation thresholds are known with six-digits accuracy or new models applying percolation theory59. Moreover, finding with better accuracy values 
, 
, and 
for simple cubic lattice and complex neighborhoods may bring a narrower spread of 
. Also, applying such checks for higher-dimensional (non-physical-dimensional) lattices (like hypercubes in four26,35,60–62 and five29,62 or higher63,64 dimensions) may be scientifically attractive.
Acknowledgements
This research was supported by a subsidy from the Polish Ministry of Science and Higher Education. We are also grateful to our anonymous reviewers for both: various valuable comments on the manuscript during the editorial process and forcing us to perform additional simulations (particularly for three-dimensional cases) and further conclusions.
Author contributions
Conceptualization: K.M., Formal analysis: K.M., M.J.K., Methodology: K.M., M.J.K., Software: K.M., M.J.K., Visualization: K.M., M.J.K., Writing – original draft: K.M., Writing – review & editing: K.M., M.J.K.
Data availability
All data generated or analyzed during this study are included in this published article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
- 1.Kesten, H. Percolation Theory for Mathematicians (Brikhauser, 1982). [Google Scholar]
- 2.Stauffer, D. & Aharony, A. Introduction to Percolation Theory 2nd ed. (Taylor and Francis, 1994). 10.1201/9781315274386. [Google Scholar]
- 3.Sahimi, M. Applications of Percolation Theory (Taylor and Francis, 1994). [Google Scholar]
- 4.Hammersley, J. M. Percolation processes: II. The connective constant, Mathematical Proceedings of the Cambridge Philosophical Society,53, 642. 10.1017/S0305004100032692 (1957).
- 5.Kesten, H. The incipient infinite cluster in two-dimensional percolation. Probab. Theory Relat. Fields73, 369. 10.1007/BF00776239 (1986). [Google Scholar]
- 6.Balankin, A. S. A survey of fractal features of Bernoulli percolation. Chaos, Solitons & Fractals184, 115044. 10.1016/j.chaos.2024.115044 (2024). [Google Scholar]
- 7.Mandelbrot, B. The Fractal Geometry of Nature, Einaudi paperbacks (Henry Holt and Company, 1983). [Google Scholar]
- 8.Peitgen, H.-O., Jürgens, H. & Saupe, D. Fractals for the Classroom. Part One: Introduction to Fractals and Chaos (Springer, 1992). 10.1007/978-1-4757-2172-0. [Google Scholar]
- 9.Peitgen, H.-O., Jürgens, H. & Saupe, D. Fractals for the Classroom (Complex Systems and Mandelbrot Set (Springer, New York, NY, Part Two, 1992). 10.1007/978-1-4612-4406-6.
- 10.Isichenko, M. B. Percolation, statistical topography, and transport in random media. Rev. Mod. Phys.64, 961. 10.1103/RevModPhys.64.961 (1992). [Google Scholar]
- 11.Li, M. & Deng, Y. Iterative site percolation on triangular lattice. Phys. Rev. Res.6, 033318. 10.1103/PhysRevResearch.6.033318 (2024). [Google Scholar]
- 12.Nienhuis, B. Coulomb gas formulation of 2D phase transition. In Phase transition and critical phenomena Vol. 11 (eds Domb, C. & Lebowitz, J. L.) (Academic Press, 1987). [Google Scholar]
- 13.Saleur, H. & Duplantier, B. Exact determination of the percolation Hull exponent in two dimensions. Phys. Rev. Lett.58, 2325. 10.1103/PhysRevLett.58.2325 (1987). [DOI] [PubMed] [Google Scholar]
- 14.Cardy, J. L. Conformal invariance, In Phase transition and critical phenomena, Vol. 11, edited by C. Domb and J. L. Lebowitz (Academic Press, 1987)
- 15.Francesco, P., Mathieu, P. & Sénéchal, D. Conformal field theory (Springer Science & Business Media, 2012). [Google Scholar]
- 16.Kager, W. & Nienhuis, B. A guide to stochastic Löwner evolution and its applications. J. Stat. Phys.115, 1149. 10.1023/B:JOSS.0000028058.87266.be (2004). [Google Scholar]
- 17.Cardy, J. SLE for theoretical physicists. Ann. Phys.318, 81. 10.1016/j.aop.2005.04.001 (2005) (special Issue). [Google Scholar]
- 18.Smirnov, S. & Werner, W. Critical exponents for two-dimensional percolation. Math. Res. Lett.8, 729–744. 10.4310/MRL.2001.v8.n6.a4 (2001). [Google Scholar]
- 19.Nienhuis, B. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys.34, 731. 10.1007/BF01009437 (1984). [Google Scholar]
- 20.Kapitulnik, A., Aharony, A., Deutscher, G. & Stauffer, D. Self similarity and correlations in percolation. J. Phys. A: Math. Gen.16, L269. 10.1088/0305-4470/16/8/003 (1983). [Google Scholar]
- 21.Dalton, N. W., Domb, C. & Sykes, M. F. Dependence of critical concentration of a dilute ferromagnet on the range of interaction. Proc. Phys. Soc.83, 496. 10.1088/0370-1328/83/3/118 (1964). [Google Scholar]
- 22.Domb, C. & Dalton, N. W. Crystal statistics with long-range forces: I. The equivalent neighbour model, Proceedings of the Physical Society89, 859. 10.1088/0370-1328/89/4/311 (1966). [Google Scholar]
- 23.d’Iribarne, C., Rasigni, M. & Rasigni, G. From lattice long-range percolation to the continuum one. Phys. Lett. A263, 65. 10.1016/S0375-9601(99)00585-X (1999). [Google Scholar]
- 24.Xun, Z., Hao, D. & Ziff, R. M. Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit. Phys. Rev. E103, 022126. 10.1103/PhysRevE.103.022126 (2021). [DOI] [PubMed] [Google Scholar]
- 25.Xun, Z., Hao, D. & Ziff, R. M. Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions. Phys. Rev. E105, 024105. 10.1103/PhysRevE.105.024105 (2022). [DOI] [PubMed] [Google Scholar]
- 26.Zhao, P., Yan, J., Xun, Z., Hao, D. & Ziff, R. M. Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods. J. Stat. Mech: Theory Exp.2022, 033202. 10.1088/1742-5468/ac52a8 (2022). [Google Scholar]
- 27.Xun, Z. & Hao, D. Monte Carlo simulation of bond percolation on square lattice with complex neighborhoods. Acta Physica Sinica71, 066401. 10.7498/aps.71.20211757 (2022) (in Chinese). [Google Scholar]
- 28.Cirigliano, L., Castellano, C. & Timár, G. Extended-range percolation in complex networks. Phys. Rev. E108, 044304. 10.1103/PhysRevE.108.044304 (2023). [DOI] [PubMed] [Google Scholar]
- 29.Xun, Z., Hao, D. & Ziff, R. M. Extended-range percolation in five dimensions (2023), arXiv:2308.15719 [cond-mat.stat-mech]
- 30.Malarz, K. & Galam, S. Square-lattice site percolation at increasing ranges of neighbor bonds. Phys. Rev. E71, 016125. 10.1103/PhysRevE.71.016125 (2005). [DOI] [PubMed] [Google Scholar]
- 31.Galam, S. & Malarz, K. Restoring site percolation on damaged square lattices. Phys. Rev. E72, 027103. 10.1103/PhysRevE.72.027103 (2005). [DOI] [PubMed] [Google Scholar]
- 32.Majewski, M. & Malarz, K. Square lattice site percolation thresholds for complex neighbourhoods. Acta Physica Polonica B38, 2191 (2007). [Google Scholar]
- 33.Kurzawski, Ł & Malarz, K. Simple cubic random-site percolation thresholds for complex neighbourhoods. Rep. Math. Phys.70, 163. 10.1016/S0034-4877(12)60036-6 (2012). [Google Scholar]
- 34.Malarz, K. Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors. Phys. Rev. E91, 043301. 10.1103/PhysRevE.91.043301 (2015). [DOI] [PubMed] [Google Scholar]
- 35.Kotwica, M., Gronek, P. & Malarz, K. Efficient space virtualisation for Hoshen-Kopelman algorithm. Int. J. Mod. Phys. C30, 1950055. 10.1142/S0129183119500554 (2019). [Google Scholar]
- 36.Malarz, K. Site percolation thresholds on triangular lattice with complex neighborhoods. Chaos30, 123123. 10.1063/5.0022336 (2020). [DOI] [PubMed] [Google Scholar]
- 37.Malarz, K. Percolation thresholds on triangular lattice for neighbourhoods containing sites up to the fifth coordination zone. Phys. Rev. E103, 052107. 10.1103/PhysRevE.103.052107 (2021). [DOI] [PubMed] [Google Scholar]
- 38.Malarz, K. Random site percolation on honeycomb lattices with complex neighborhoods. Chaos32, 083123. 10.1063/5.0099066 (2022). [DOI] [PubMed] [Google Scholar]
- 39.Malarz, K. Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone. XXPhys. A632, 129347. 10.1016/j.physa.2023.129347 (2023). [Google Scholar]
- 40.Malarz, K. Universality of percolation thresholds for two-dimensional complex non-compact neighborhoods. Phys. Rev. E109, 034108. 10.1103/PhysRevE.109.034108 (2024). [DOI] [PubMed] [Google Scholar]
- 41.Ciepłucha, A. P., Utnicki, M., Wołoszyn, M. & Malarz, K. Lower limit of percolation threshold on a square lattice with complex neighborhoods. Entropy27, 361. 10.3390/e27040361 (2025). [DOI] [PMC free article] [PubMed] [Google Scholar]
- 42.See Supplemental Material at 10.58032/AGH/YHDDNX
-
43.Jacobsen, J. L. Critical points of Potts and
models from eigenvalue identities in periodic Temperley-Lieb algebras. J. Phys. A: Math. Theor.48, 454003. 10.1088/1751-8113/48/45/454003 (2015). [Google Scholar] - 44.Xun, Z., Hao, D. & Ziff, R. M. Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit. Phys. Rev. E103, 022126. 10.1103/PhysRevE.103.022126 (2021). [DOI] [PubMed] [Google Scholar]
- 45.Newman, M. E. J. & Ziff, R. M. Fast Monte Carlo algorithm for site or bond percolation. Phys. Rev. E64, 016706. 10.1103/PhysRevE.64.016706 (2001). [DOI] [PubMed] [Google Scholar]
- 46.Privman, V. Finite-size scaling theory, In Finite size scaling and numerical simulation of statistical systems, edited by V. Privman (World Scientific, Singapore, 1990) pp. 1–98 10.1142/9789814503419_0001
- 47.Landau, D. P. & Binder, K. A Guide to Monte Carlo Simulations in Statistical Physics 3rd ed. (Cambridge University Press, 2009). [Google Scholar]
- 48.Fayfar, S., Bretaña, A. & Montfrooij, W. Protected percolation: A new universality class pertaining to heavily-doped quantum critical systems. J. Phys. Commun.5, 015008. 10.1088/2399-6528/abd8e9 (2021). [Google Scholar]
- 49.Brzeski, P. & Kondrat, G. Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous. J. Stat. Mech: Theory Exp.2022, 053202. 10.1088/1742-5468/ac6519 (2022). [Google Scholar]
- 50.Hoshen, J. & Kopelman, R. Percolation and cluster distribution. 1. Cluster multiple labeling technique and critical concentration algorithm. Phys. Rev. B14, 3438. 10.1103/PhysRevB.14.3438 (1976). [Google Scholar]
- 51.Hoshen, J. & Kopelman, R. Exciton percolation. 1. Migration dynamics. J. Chem. Phys.2817, 65. 10.1063/1.433430 (1976). [Google Scholar]
- 52.Deng, Y., Ouyang, Y. & Blöte, H. W. J. Medium-range percolation in two dimensions. J. Phys: Conf. Ser.1163, 012001. 10.1088/1742-6596/1163/1/012001 (2019). [Google Scholar]
- 53.Xu, X., Wang, J., Lv, J.-P. & Deng, Y. Simultaneous analysis of three-dimensional percolation models. Front. Phys.9, 113. 10.1007/s11467-013-0403-z (2014). [Google Scholar]
- 54.Abramowitz, M. & Stegun, I. A. (eds)., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 6th ed. (United States. Government Printing Office, Washington D.C., 1964)
- 55.Lorenz, C. D. & Ziff, R. M. Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices. Phys. Rev. E57, 230. 10.1103/PhysRevE.57.230 (1998). [Google Scholar]
- 56.Jan, N. & Stauffer, D. Random site percolation in three dimensions. Int. J. Mod. Phys. C09, 341. 10.1142/S0129183198000261 (1998). [Google Scholar]
- 57.Ballesteros, H. G. et al. Scaling corrections: site percolation and Ising model in three dimensions. J. Phys. A: Math. Gen.32, 1. 10.1088/0305-4470/32/1/004 (1999). [Google Scholar]
- 58.Deng, Y. & Blöte, H. W. J. Monte Carlo study of the site-percolation model in two and three dimensions. Phys. Rev. E72, 016126. 10.1103/PhysRevE.72.016126 (2005). [DOI] [PubMed] [Google Scholar]
- 59.Das, J. R., Sinha, S., Hansen, A. & Santra, S. B. Mixed-wet percolation on a dual square lattice (2025), arXiv:2503.06336 [cond-mat.stat-mech]. 10.48550/arXiv.2503.06336
- 60.Xun, Z. & Ziff, R. M. Precise bond percolation thresholds on several four-dimensional lattices. Phys. Rev. Res.2, 013067. 10.1103/PhysRevResearch.2.013067 (2020). [Google Scholar]
- 61.Ballesteros, H. G. et al. Measures of critical exponents in the four-dimensional site percolation. Phys. Lett. B400, 346. 10.1016/S0370-2693(97)00337-7 (1997). [Google Scholar]
- 62.Paul, G., Ziff, R. M. & Stanley, H. E. Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions. Phys. Rev. E64, 026115. 10.1103/PhysRevE.64.026115 (2001). [DOI] [PubMed] [Google Scholar]
- 63.Mertens, S. & Moore, C. Percolation thresholds and Fisher exponents in hypercubic lattices. Phys. Rev. E98, 022120. 10.1103/PhysRevE.98.022120 (2018). [DOI] [PubMed] [Google Scholar]
-
64.Hu, Y. & Charbonneau, P. Percolation thresholds on high-dimensional
and 
-related lattices. Phys. Rev. E103, 062115. 10.1103/PhysRevE.103.062115 (2021).
[DOI] [PubMed] [Google Scholar]
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All data generated or analyzed during this study are included in this published article.