# Simultaneous analysis of three-dimensional percolation models

Xiao Xu  Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Junfeng Wang  Hefei National Laboratory for
Physical Sciences at Microscale and Department of Modern Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  School of
Electronic Science and Applied Physics, Hefei University of Technology, Hefei,
Anhui 230009, China  Jian-Ping Lv<EMAIL_ADDRESS>Department of Physics,
China University of Mining and Technology, Xuzhou 221116, China  Youjin Deng
<EMAIL_ADDRESS>Hefei National Laboratory for Physical Sciences at
Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China

###### Abstract

We simulate the bond and site percolation models on several three-dimensional
lattices, including the diamond, body-centered cubic, and face-centered cubic
lattices. As on the simple-cubic lattice [Phys. Rev. E, 87 052107 (2013)], it
is observed that in comparison with dimensionless ratios based on cluster-size
distribution, certain wrapping probabilities exhibit weaker finite-size
corrections and are more sensitive to the deviation from percolation threshold
$p_{c}$, and thus provide a powerful means for determining $p_{c}$. We analyze
the numerical data of the wrapping probabilities simultaneously such that
universal parameters are shared by the aforementioned models, and thus
significantly improved estimates of $p_{c}$ are obtained.

###### pacs:

05.50.+q (lattice theory and statistics), 05.70.Jk (critical point phenomena),
64.60.ah (percolation), 64.60.F- (equilibrium properties near critical points,
critical exponents)

##  I Introduction

Percolation is a geometric model which involves the random occupation of sites
or edges of a regular lattice, and was first introduced by Broadbent and
Harmmersley Broadbent and Hammersley (1957). As a cornerstone of the theory of
critical phenomena Stauffer and Aharony (1994) and a central topic in
probability theory Grimmett (1999); Bollobás and Riordan (2006), percolation
attracts much attention.

The two-dimensional (2D) case has been studied extensively, and several exact
results are known. Coulomb gas arguments Nienhuis (1987) and conformal field
theory Cardy (1987) predict the exact values of the bulk critical exponents
$\beta=5/36$ and $\nu=4/3$, which have been confirmed rigorously in the
specific case of site percolation on the triangular lattice Smirnov and Werner
(2001). Moreover, percolation thresholds $p_{c}$ on many 2D lattices are
exactly known Essam (1972), or known to very high precision Feng et al.
(2009); Ding (2010). For $d>2$, estimates of $p_{c}$ have to rely on numerical
methods such as series expansions and Monte Carlo simulations, while the
critical exponents $\beta=1$ and $d\nu=3$ for $d\geq d_{c}=6$ can be predicted
by mean-field theory Toulouse (1974) and even proved rigorously Aizenman and
Newman (1984); Hara and Slade (1990) for $d\geq 19$. A more or less thorough
list of percolation thresholds for $d\in[2,13]$ is summarized on the Wikipedia
webpage: http://en.wikipedia.org/wiki/Percolation_threshold.

Very recently, two of the authors and coworkers carried out an extensive
simulation of bond and site percolation on the simple-cubic (SC) lattice up to
system size $512\times 512\times 512$ Wang et al. (2013), and determined the
percolation thresholds and critical exponents to high precision. It was
observed that in comparison with dimensionless ratios based on cluster-size
moments, the wrapping probabilities suffer from weaker finite-size corrections
and are more sensitive to the deviation $p-p_{c}$ from the percolation
threshold. As an extension of Ref. Wang et al. (2013), the present work
studies percolation on other common three-dimensional (3D) lattices, and shows
that such an observation generally holds in 3D percolation. Meanwhile, with
the employment of a simultaneous fitting procedure developed in Ref. Deng and
Blöte (2003) and the help of the accurate data reported in Ref. Wang et al.
(2013), we also provide high-precision estimates of $p_{c}$ for the site and
bond percolation on the diamond (DM), body-centered cubic (BCC), and face-
centered cubic (FCC) lattices.

The remainder of this paper is organized as follows. Section II defines the
sampled quantities of interest. In Sec. III, the numerical data of the
dimensionless ratios and the wrapping probabilities are analyzed separately
for each percolation model. Then, a simultaneous fitting of the wrapping
probabilities is carried out to determine percolation threshold $p_{c}$.
Section IV presents the analyses for other quantities at criticality $p_{c}$,
and a brief discussion is given in Sec. V.

##  II Sampled quantities

We study bond and site percolation on three-dimensional lattices including the
DM, SC, BCC, and FCC lattices, illustrated in Fig. 1. The simulations follow
the standard method: each edge/site is occupied with probability $p$ and
clusters are constructed by the breadth-first search.

Figure 1: Three-dimensional lattices: (left-top), SC; (righttop), DM; (left-
bottom), BCC; (right-bottom), FCC.

The sampled quantities are the same as in Ref. Wang et al. (2013). For
completeness, they are described in the following.

  * •

The number of occupied bonds ${\mathcal{N}}_{b}$ or sites ${\mathcal{N}}_{s}$.

  * •

The number of clusters ${\mathcal{N}}_{c}$.

  * •

The largest-cluster size ${\mathcal{C}}_{1}$.

  * •

The cluster-size moments ${\mathcal{S}}_{m}=\sum_{C}|C|^{m}$ with $m=2,4$,
where the sum runs over all clusters $C$ and $|C|$ denotes cluster size.

  * •

An observable ${\mathcal{S}}:=\max\limits_{C}\,\max\limits_{y\in
C}\,d(x_{C},y)$ used to determine the shortest-path exponent. Here $d(x,y)$
denotes the graph distance from vertex $x$ to vertex $y$, and $x_{C}$ is the
vertex in cluster $C$ with the smallest vertex label, according to some fixed
(but arbitrary) vertex labeling.

  * •

The indicators ${\mathcal{R}}^{(x)}$, ${\mathcal{R}}^{(y)}$, and
${\mathcal{R}}^{(z)}$, for the event that a cluster wraps around the lattice
in the $x$, $y$, or $z$ directions, respectively.

From these observables we calculated the following quantities:

  * •

The mean size of the largest cluster $C_{1}=\langle{\mathcal{C}}_{1}\rangle$,
which scales as $C_{1}\sim L^{y_{h}}$ at $p_{c}$, with $L$ the linear system
size and $y_{h}=d-\beta/\nu$.

  * •

The cluster density $\rho=\langle{\mathcal{N}}_{c}\rangle/V$, where $V=gL^{3}$
is the number of lattice sites, with $g=1$ for the SC and DM lattices, $g=2$
for the BCC lattice, and $g=4$ for the FCC lattice.

  * •

The dimensionless ratios

$Q_{1}=\frac{\langle{{\mathcal{C}}_{1}}^{2}\rangle}{\langle{\mathcal{C}}_{1}\rangle^{2}}\;,\;\;\;Q_{2}=\frac{\langle{{\mathcal{S}}_{2}}^{2}\rangle}{\langle
3{{\mathcal{S}}_{2}}^{2}-2{\mathcal{S}}_{4}\rangle}\;.$ (1)

In the case of the Ising model, $Q_{2}$ is identical to the dimensionless
ratio $Q_{M}=\langle M^{2}\rangle^{2}/\langle M^{4}\rangle$, where $M$
represents the magnetization.

  * •

The mean shortest-path length $S=\langle{\mathcal{S}}\rangle$, which at
$p_{c}$ scales like $S\sim L^{d_{\rm min}}$ with $d_{\rm min}$ the shortest-
path fractal dimension.

  * •

The wrapping probabilities

$\displaystyle R^{(x)}=$
$\displaystyle\langle{\mathcal{R}}^{(x)}\rangle=\langle{\mathcal{R}}^{(y)}\rangle=\langle{\mathcal{R}}^{(z)}\rangle\;,$
(2) $\displaystyle R^{(a)}=$ $\displaystyle
1-\langle(1-{\mathcal{R}}^{(x)})(1-{\mathcal{R}}^{(y)})(1-{\mathcal{R}}^{(z)})\rangle\;,$
$\displaystyle R^{(3)}=$
$\displaystyle\langle{\mathcal{R}}^{(x)}{\mathcal{R}}^{(y)}{\mathcal{R}}^{(z)}\rangle\;.$

Here $R^{(x)}$, $R^{(a)}$ and $R^{(3)}$ give the probability that a winding
exists in the $x$ direction, in at least one of the three possible directions,
and simultaneously in the three directions, respectively. At $p_{c}$, these
wrapping probabilities take non-zero universal values in the thermodynamic
limit $L\rightarrow\infty$.

  * •

The covariance of ${\mathcal{R}}^{(x)}$ and ${\mathcal{N}}_{b}$

$g^{(x)}_{bR}=\langle{\mathcal{R}}^{(x)}{\mathcal{N}}_{b}\rangle-\langle{\mathcal{R}}^{(x)}\rangle\langle{\mathcal{N}}_{b}\rangle\;,$
(3)

which scales as $g^{(x)}_{bR}\sim L^{y_{t}}=L^{1/\nu}$ at criticality $p_{c}$.
Analogously, one defines $g^{(x)}_{sR}$ for site percolation, with
${\mathcal{N}}_{b}$ being replaced with ${\mathcal{N}}_{s}$.

Figure 2: Quantities $Q_{2}$ and $R^{(x)}$ as a function of $p$ for the site
percolation on the DM lattice with various sizes. In comparison with $Q_{2}$,
the plot of $R^{(x)}$ has a finer vertical scale, but still displays a clearer
intersection. This suggests that $R^{(x)}$ suffers weaker finite-size
corrections and provides a better estimator for $p_{c}$.

##  III Percolation threshold

The simulation on the SC lattice is up to linear size $L_{\rm max}=512$, and
the number of samples is about $5\times 10^{8}$ for $L\leq 128$, $6\times
10^{7}$ for $L=256$, and $3\times 10^{7}$ for $L=512$. The Monte Carlo data
and the analysis have been reported in Ref. Wang et al. (2013). For the other
lattices, the simulation is less extensive with $L_{\rm max}=128$. The number
of samples is about $10^{8}$ for lattice $L<128$ and $4\times 10^{7}$ for
$L=128$.

Table 1: Percolation thresholds from the separate fits of the wrapping
probabilities and the dimensionless ratios.

| $Q_{1}$ | $Q_{2}$ | $R^{(x)}$ | $R^{(a)}$ | $R^{(3)}$
---|---|---|---|---|---
$\rm{DM}^{b}$ | 0.389 591(2) | 0.389 592(1) | 0.389 589 2(5) | 0.389 588 9(4) | 0.389 590 0(5)
$\rm{DM}^{s}$ | 0.429 987(2) | 0.429 985(1) | 0.429 987 7(9) | 0.429 987 5(6) | 0.429 987 3(4)
$\rm{SC}^{b}$ | 0.248 811 96(6) | 0.248 811 92(6) | 0.248 811 85(3) | 0.248 811 80(4) | 0.248 811 81(9)
$\rm{SC}^{s}$ | 0.311 606 9(2) | 0.311 607 1(2) | 0.311 607 68(7) | 0.311 607 74(6) | 0.311 607 7(1)
$\rm{BCC}^{b}$ | 0.180 287 8(9) | 0.180 288 3(6) | 0.180 287 5(2) | 0.180 287 4(2) | 0.180 287 9(2)
$\rm{BCC}^{s}$ | 0.245 961 7(3) | 0.245 961 5(2) | 0.245 961 7(2) | 0.245 961 70(11) | 0.245 961 7(3)
$\rm{FCC}^{b}$ | 0.120 163 9(5) | 0.120 163 3(3) | 0.120 163 6(2) | 0.120 163 6(2) | 0.120 163 7(3)
$\rm{FCC}^{s}$ | 0.199 235 3(3) | 0.199 235 2(2) | 0.199 235 2(2) | 0.199 235 14(11) | 0.199 235 0(2)

Table 2: Value of the amplitudes $q_{1}$ obtained from the separate fits of
the wrapping probabilities and the dimensionless ratios.

| $Q_{1}$ | $Q_{2}$ | $R^{(x)}$ | $R^{(a)}$ | $R^{(3)}$
---|---|---|---|---|---
$\rm{DM}^{b}$ | 0.277(2) | 0.642(3) | 0.906(4) | 1.236(5) | 0.484(3)
$\rm{DM}^{s}$ | 0.193(2) | 0.458(4) | 0.652(3) | 0.894(4) | 0.341(3)
$\rm{SC}^{b}$ | 0.30(3) | 0.90(7) | 1.20(7) | 1.80(9) | 0.65(7)
$\rm{SC}^{s}$ | 0.22(2) | 0.52(4) | 0.70(4) | 1.00(3) | 0.36(3)
$\rm{BCC}^{b}$ | 0.644(3) | 1.46(2) | 2.084(8) | 2.82(3) | 1.12(2)
$\rm{BCC}^{s}$ | 0.30(1) | 0.72(3) | 1.04(2) | 1.42(2) | 0.56(1)
$\rm{FCC}^{b}$ | 1.19(9) | 2.77(4) | 3.91(3) | 5.29(2) | 2.08(2)
$\rm{FCC}^{s}$ | 0.449(3) | 1.044(9) | 1.507(5) | 2.080(6) | 0.794(4)

Table 3: Percolation thresholds and other non-universal parameters from the
simultaneous fits of the wrapping probabilities. For all the fits, we set
$L_{\rm min}=32$ for $R^{(x)}$ and $R^{(3)}$ and $L_{\rm min}=24$ for
$R^{(a)}$, $a$, $b_{1}$ and $b_{2}$ are defined in Eq. (5).

M. | Obs. | $p_{c}$ | $a$ | $b_{1}$ | $b_{2}$ | M. | Obs. | $p_{c}$ | $a$ | $b_{1}$ | $b_{2}$
---|---|---|---|---|---|---|---|---|---|---|---
$\rm{DM}^{b}$ | $R^{(x)}$ | 0.389 589 22(18) | $0.901(4)$ | $0.012(13)$ | $0.04(17)$ | $\rm{DM}^{s}$ | $R^{(x)}$ | 0.429 986 96(19) | $0.653(4)$ | $0.023(9)$ | $-0.53(11)$
$R^{(a)}$ | 0.389 589 1(1) | $1.236(2)$ | $-0.006(6)$ | $0.08(7)$ | $R^{(a)}$ | 0.429 987 15(12) | $0.895(2)$ | $0.043(5)$ | $-0.73(6)$
$R^{(3)}$ | 0.389 589 40(20) | $0.480(1)$ | $0.011(7)$ | $0.1(1)$ | $R^{(3)}$ | 0.429 986 81(24) | $0.3463(8)$ | $-0.001(6)$ | $-0.27(7)$
$\rm{SC}^{b}$ | $R^{(x)}$ | 0.248 811 84(3) | $1.25(2)$ | $0.001(5)$ | $0.29(6)$ | $\rm{SC}^{s}$ | $R^{(x)}$ | 0.311 607 65(5) | $0.721(5)$ | $0.024(4)$ | $-0.44(5)$
$R^{(a)}$ | 0.248 811 85(3) | $1.69(1)$ | $-0.011(4)$ | $0.78(4)$ | $R^{(a)}$ | 0.311 607 69(4) | $0.992(4)$ | $0.036(3)$ | $0.02(3)$
$R^{(3)}$ | 0.248 811 94(5) | $0.651(8)$ | $0.004(4)$ | $0.03(5)$ | $R^{(3)}$ | 0.311 607 70(8) | $0.384(4)$ | $0.002(4)$ | $-0.46(4)$
$\rm{BCC}^{b}$ | $R^{(x)}$ | 0.180 287 6(1) | $2.069(8)$ | $-0.006(7)$ | $0.2(1)$ | $\rm{BCC}^{s}$ | $R^{(x)}$ | 0.245 961 48(6) | $1.032(7)$ | $0.020(4)$ | $-0.41(5)$
$R^{(a)}$ | 0.180 287 57(9) | $2.839(4)$ | $-0.016(4)$ | $~{}~{}0.01(5)$ | $R^{(a)}$ | 0.245 961 51(6) | $1.407(3)$ | $0.027(3)$ | $-0.46(3)$
$R^{(3)}$ | 0.180 287 65(9) | $1.102(3)$ | $-0.004(4)$ | $0.27(6)$ | $R^{(3)}$ | 0.245 961 46(9) | $0.543(2)$ | 0.001(3) | $-0.19(4)$
$\rm{FCC}^{b}$ | $R^{(x)}$ | 0.120 163 79(7) | $3.87(2)$ | $0.004(8)$ | $0.05(12)$ | $\rm{FCC}^{s}$ | $R^{(x)}$ | 0.199 235 17(6) | $1.48(3)$ | $0.011(6)$ | $-0.13(8)$
$R^{(a)}$ | 0.120 163 80(5) | $5.311(6)$ | $-0.008(4)$ | $0.04(5)$ | $R^{(a)}$ | 0.199 235 22(5) | $2.077(3)$ | $0.018(4)$ | $-0.13(4)$
$R^{(3)}$ | 0.120 163 72(18) | $2.059(5)$ | $0.014(6)$ | $-0.1(1)$ | $R^{(3)}$ | 0.199 235 12(9) | $0.804(2)$ | $0.002(5)$ | $0.09(6)$

###  III.1 Separate fits

In numerical study of phase transitions, dimensionless ratios like $Q_{1}$ and
$Q_{2}$ are known to provide powerful tools for locating critical points
$p_{c}$. The wrapping probabilities have analogous finite-size scaling
behaviors as the dimensionless ratios, and thus should also provide a useful
method for estimating $p_{c}$. This is demonstrated in Fig. 2 for site
percolation on the DM lattice. The intersections of the $Q_{2}$ data for
different sizes $L$ would approximately give the percolation threshold
$p_{c}\approx 0.429\,95$, with uncertainty at the fourth or fifth decimal
place. Due to their faster convergence as $L$ increases, the intersections of
the $R^{(x)}$ data would yield $p_{c}\approx 0.429\,99$. Similar phenomena are
observed in all the percolation models studied in this work. Thus, it clearly
suggests that the wrapping probabilities are more powerful tools for
estimating $p_{c}$ than the dimensionless ratios $Q_{1}$ and $Q_{2}$.

According to the least-squares criterion, we fit Monte Carlo data for the
quantities $R^{(x)}$, $R^{(a)}$, $R^{(3)}$, $Q_{1}$ and $Q_{2}$ separately for
each percolation model to the following scaling ansatz

$\displaystyle U(p,L)$ $\displaystyle=$ $\displaystyle
U_{0}+\sum_{k=1}^{3}q_{k}(p-p_{c})^{k}L^{ky_{t}}$ (4)
$\displaystyle+b_{1}L^{-1.2}+b_{2}L^{-2}\;,$

where $y_{t}$ is the thermal exponent, $U_{0}$ is a universal value depending
on the quantity studied, and the $q_{k}$ ($k=1,2,3$) and $b_{j}$ ($j=1,2$) are
non-universal constants. A correction exponent of $-1.2$ is taken from the
existing literature Wang et al. (2013). To evaluate the systematic errors
caused by the scaling terms which are not included in the fitting ansatz, we
set a lower cutoff $L\geq L_{\rm min}$ on the data and study the effect on the
residual $\chi^{2}$ as $L_{\rm min}$ increases. Generally, we prefer the fit
which produces $\chi^{2}/DF\sim O(1)$ ($DF$ is the degree of freedom), and in
which the subsequent increases of $L_{\rm min}$ do not drop $\chi^{2}$ by
vastly more than one unit per degree of freedom. These principles apply in all
the fits we carry out.

In the fits, we try different combinations of corrections to scaling: (1) both
$b_{1}$ and $b_{2}$ are free to be determined by the data; (2), $b_{1}$ is set
to $0$ and $b_{2}$ is free; and (3), $b_{1}$ is free and $b_{2}$ is fixed at
$0$. We find that the correction amplitudes $b_{1}$ for the wrapping
probabilities are rather small and in many cases are statistically consistent
with zero. In contrast, for the dimensionless ratios one clearly observes a
non-zero correction amplitude $b_{1}$. Moreover, the amplitudes $q_{1}$ of the
term $q_{1}(p-p_{c})L^{y_{t}}$ in Eq. (4) for $R^{(x)}$ and $R^{(a)}$ are
larger than those for $Q_{1}$ and $Q_{2}$. This suggests that the wrapping
probabilities are more sensitive to the deviation from criticality $p-p_{c}$
than the dimensionless ratios. These observations in the fits are reflected by
Fig. 2.

Tables 1 and 2 summarize the percolation thresholds and the amplitudes $q_{1}$
from our preferred fits with combination (1), where the uncertainties are just
the statistical errors. It can be seen that the estimates of $p_{c}$ from
different quantities are consistent with each other within the combined error
margins. Further, the wrapping probabilities yield more accurate estimate of
$p_{c}$ than the dimensionless ratios by a factor of two or three.

Table 4: Simultaneous fits of the wrapping probabilities
$R^{(x)},R^{(a)},R^{(3)}$ for all models.

Obs. | $y_{t}$ | $U_{0}$ | $U_{2}$ | $U_{3}$
---|---|---|---|---
$R^{(x)}$ | 1.1424(11) | $0.257\,80(6)$ | $1.23(1)$ | $-0.9(6)$
$R^{(a)}$ | 1.1418(4) | $0.460\,02(2)$ | $0.311(2)$ | $-0.99(4)$
$R^{(3)}$ | 1.1413(6) | $0.080\,46(4)$ | $4.98(1)$ | $9.7(4)$

Figure 3: $R^{(x)}(p,L)$ versus $L^{y_{t}}$ at given $p$ values which are in close to the estimated percolation thresholds for the site and bond percolation on the BCC (top), FCC (middle) and DM (bottom) respectively. Table 5: Final estimates of percolation thresholds for the three-dimensional percolation models. The error bars include both statistical and systematic errors.  Lattice | Bond |  |  | Site |
---|---|---|---|---|---
|  $p_{c}$(Present) |  $p_{c}$(Previous) |  |  $p_{c}$(Present) |  $p_{c}$(Previous)
DM | 0.389 589 2(5) | 0.389 3(2) Marck (1998) |  | 0.429 987 0(4) | 0.430 1(4) Marck (1998)
|  | 0.390(11) Vyssotsky et al. (1961) |  |  | 0.426(+0.08,-0.02) Silverman and Adler (1990)
SC | 0.248 811 85(10) | 0.248 811 82(10) Wang et al. (2013) |  | 0.311 607 68(15) | 0.311 607 7(2) Wang et al. (2013)
|  | 0.248 812 6(5) Lorenz and Ziff (1998b) |  |  | 0.311 607 4(4) Deng and Blöte (2005)
BCC | 0.180 287 62(20) | 0.180 287 5(10) Lorenz and Ziff (1998a) |  | 0.245 961 5(2) | 0.245 961 5(10) Lorenz and Ziff (1998b)
|  |  |  |  | 0.246 0(3) Bradley et al. (1991), 0.246 4(7) Gaunt and Sykes (1983)
FCC | 0.120 163 77(15) | 0.120 163 5(10) Lorenz and Ziff (1998a) |  | 0.199 235 17(20) | 0.199 236 5(10) Lorenz and Ziff (1998b)

###  III.2 Simultaneous fits

As described above, the Monte Carlo simulations for the SC lattice are much
more extensive and are performed on larger system sizes than those on the
other lattices. This leads to the more precise estimates of $p_{c}$ and other
parameters on the SC lattices. It is noted that for a given wrapping
probability or dimensionless ratio, the value of $U_{0}$ in Eq. (4) is
universal. To make use of the extensive simulation for the SC lattice, we
carry out a simultaneous analysis of the Monte Carlo data for all the
percolation systems studied in this work. More precisely, we choose the
wrapping probabilities $R^{(x)}$, $R^{(a)}$, and $R^{(3)}$, and for each of
them, the data is fitted by

$\displaystyle U(p_{j},L)$ $\displaystyle=$ $\displaystyle
U_{0}+\sum_{k=1}^{3}U_{k}a_{j}^{k}(p_{j}-p_{c,j})^{k}L^{ky_{t}}$ (5)
$\displaystyle+b_{1,j}L^{-1.2}+b_{2,j}L^{-2}\;,$

where $U_{k}\;(k=0,1,2,3)$ and $y_{t}$ are universal; $j\in\\{1,2,...,8\\}$
refer to the site and bond percolation models on DM, SC, BCC and FCC lattice,
and the parameters with subscript $j$ are model-dependent. In other words, Eq.
(5) can be regarded as a set of equations in which the universal parameters
$U_{k}$ and $y_{t}$ are shared by all the percolation models. We expect that
an accurate estimation of these universal parameters will be mainly achieved
by the high-precision Monte Carlo data on the SC lattice, and as in return,
this will help to improve the accuracy of $p_{c}$ for the other models. Such a
simultaneous analysis has been applied to the 3D Ising model, and the
derivation of Eq. (5) can be found in Ref. Deng and Blöte (2003).

The simultaneous fits by Eq. (5) follow the same procedure as that in the
above subsection. We first note that among $U_{1}$ and $a_{j}$ with
$j=1,\cdots,8$, there is one redundant parameter, and we thus set $U_{1}=1$.

Tables 3 and 4 summarize the results for the universal parameters, the
percolation thresholds, and other non-universal constants, taken from the
preferred fits with $L_{\rm min}=24$ or $32$. In these fits, both the
correction amplitudes $b_{1,j}$ and $b_{2,j}$ are left free. It can be seen
from Tab. 3 that the leading correction amplitudes $b_{1,j}$ are rather small.
In the cases that $b_{1,j}$ cannot be distinguished from zero within the
statistical uncertainties, one can in principle exclude the leading correction
term in the fits, which will further decrease the error margins.

In comparison with the results in Tab. 1 from the separate fits, the
simultaneous analyses do significantly improve the estimates of $p_{c}$. By
taking into account the results from different wrapping probabilities and from
fits with different $L_{\rm min}$, we obtain the final estimates of $p_{c}$,
as summarized in Tab. 5. To check the reliability of the final quoted error
margins in Tab. 5, we plot the $R^{(x)}$ data at $p_{c}$ and two other $p$
values which are away from $p_{c}$ about four or five times of final error
bars. Precisely at $p=p_{c}$, the $R^{(x)}$ data should tend to a horizontal
line as $L\rightarrow\infty$, whereas the data at $p\neq p_{c}$ will bend
upward or downward. This is indeed clearly seen in these plots, some of which
are shown in Fig. 3, confirming the reliability of our final results in Tab.
5.

Also presented in Tab. 5 are existing estimates of $p_{c}$ from the
literature. It can be seen that this work does provide the percolation
thresholds $p_{c}$ with higher precision. For the bond and site percolation
models on the DM lattices, such improvement is significant.

##  IV Results at $p_{c}$

By fixing $p$ at or very close to the estimated thresholds $p_{c}$ in Tab. 5,
we study the covariances $g^{(x)}_{bR}$ and $g^{(x)}_{sR}$, the largest-
cluster size $C_{1}$, the shortest-path length $S$, and the cluster-number
density $\rho$. From their finite-size-scaling behaviors, one can determine
the thermal and magnetic renormalization exponent $y_{t}$ and $y_{h}$, the
shortest-path fractal dimension $d_{\rm min}$, and the universal excess
cluster number $b$. In addition, we also obtain the thermodynamic cluster-
number densities $\rho_{c}$ for the studied percolation models.

Figure 4: Log-log plot of $C_{1}$ , $g^{(x)}_{b(s)R}$ and $S$ versus the
rescaled linear size $L^{*}$ for all the $8$ percolation models. We set
$L=L^{*}$ for the bond percolation on the SC lattice, and rescale $L$ by a
constant factor (model-dependent) to collapse the numerical data.

###  IV.1 Exponents $y_{t}$, $y_{h}$ and $d_{\rm min}$

Following an analogous simultaneous analysis procedure, we fit the data of
$g^{(x)}_{bR}$ and $g^{(x)}_{sR}$, $C_{1}$, and $S$ by the ansatz

${\mathcal{A}}=L^{y_{\mathcal{A}}}(a_{0,j}+b_{1,j}L^{-1.2}+b_{2,j}L^{-2})\;,$
(6)

where ${y_{\mathcal{A}}}$ is the universal scaling exponent. It is $y_{t}$ for
covariance $g^{(x)}_{bR}$ and $g^{(x)}_{sR}$, $y_{h}$ for the largest-cluster
size $C_{1}$, and $d_{\rm min}$ for the shortest-path length $S$. We obtain
$y_{t}=1.141\,3(15)$, $y_{h}=2.522\,93(10)$, and $d_{\rm min}=1.375\,5(3)$,
which are consistent with the estimates in Ref. Wang et al. (2013), with
comparable or slightly better precision. For an illustration of these
universal exponents, we plot in the log-log scale the data of these quantities
versus the rescaled linear size $L^{*}=wL$, with constant $w=1$ for the bond
percolation on the SC lattice.

###  IV.2 Excess number of clusters

The numerical data of the cluster-number density at percolation thershold for
all the studied percolation models are simultaneous fitted by the scaling
ansatz

$\displaystyle\rho=\rho_{c}+V^{-1}(b+b_{1,j}L^{-2})\;,$ (7)

where $V$ is the number of lattice sites, and the correction amplitude $b$ is
known to be also universal and is referred to as the excess cluster number in
Ref. Ziff et al. (1997). The subleading correction is taken to be $-2$. Due to
the rapid decay of the correction term, the finite-$L$ data of $\rho$ quickly
converges to the thermodynamic value $\rho_{c}$; the well-determined values of
$\rho_{c}$ then aids in estimating the correction amplitude $b$ from the
small-$L$ data. The fitting results of $\rho_{c}$ and $b$ are shown in Table
6. Taking into account some potential systematic errors–e.g., due to the small
deviation of the simulated $p$ value from $p_{c}$, we have the final estimate
$b=0.675(1)$.

Table 6: Simultaneous fits of $\rho$ at the thresholds. Fitting parameter
$L_{\rm min}=16$ is set for all the models.

M. | $\rho_{c}$ | $b$ | $b_{1}$
---|---|---|---
$\rm DM^{b}$ | 0.231 953 78(4) | $0.674\,7(4)$ | $-0.6(5)$
$\rm DM^{s}$ | 0.075 519 45(2) | $-1.1(4)$
$\rm SC^{b}$ | 0.272 932 836(9) | 1.1(2)
$\rm SC^{s}$ | 0.052 438 217(3) | $-0.02(12)$
$\rm BCC^{b}$ | 0.298 343 834(12) | $0.3(3)$
$\rm BCC^{s}$ | 0.040 045 144(3) | $-0.76(9)$
$\rm FCC^{b}$ | 0.307 691 25(2) | $0.1(2)$
$\rm FCC^{s}$ | 0.026 526 453(4) | $-0.3(2)$

Figure 5:  Excess cluster number $V(\rho-\rho_{c})$ ($\equiv b$) versus
$L^{-1}$(left) and $L^{3}$(right) for SC site (top) and BCC site (bottom)
percolation models. The dashed straight lines represent constant $0.675$.

An illustration of the excess cluster number $b$ is shown in Fig. 5, where the
values of $V(\rho-\rho_{c})$ are plotted versus $1/L$ for the site percolation
on the SC and the BCC lattices. It can be seen that the $V(\rho-\rho_{c})$
values at $p=p_{c}$ quickly converge to $b=0.675$, while those for $p\neq
p_{c}$ are either bending downward or upward. However, this does not imply
that the cluster-number density $\rho$ provides a good quantity for locating
$p_{c}$. Near $p_{c}$, the finite-size behavior of $\rho(p,L)$ near threshold
$p_{c}$ can be described by

$\displaystyle\rho(p,L)$ $\displaystyle=$
$\displaystyle\rho_{c}+f_{1}(p-p_{c})+f_{2}(p-p_{c})^{2}+V^{-1}[b+$ (8)
$\displaystyle
h_{1}(p-p_{c})L^{y_{t}}+h_{2}(p-p_{c})^{2}L^{2y_{t}}+\cdots]\;,$

where $f_{i}$ and $h_{i}$ ($i=1,2$) are non-universal parameters. The critical
density $\rho_{c}$ and the terms with $f_{i}$ arise from the analytical part
of $\rho(p,L)$ and do not depend on size $L$. They dominate the finite-size
scaling of $\rho(p,L)$ but cannot be used to determine $p_{c}$. This is
illustrated in Fig. 5. The critical singularity is reflected in the subleading
terms with $L$-dependence. For the site percolation on the SC lattice, the fit
yields $p_{c}=0.311\,604(2)$, with much larger error margin than those from
wrapping probabilities.

##  V Summary

We present a Monte Carlo study of the bond and site percolation on several
three-dimensional lattices, and obtain high-precision estimates of the
percolation thresholds (Tab. 5), the cluster density (Tab. 6), the wrapping
probabilities (Tab. 4) and the excess cluster number b = 0.675(1). These
accurate scientific data can serve as a testing ground for future study of
systems in the percolation universality class. More importantly, it is
observed that the wrapping probabilities can be a useful and reliable approach
for locating phase transitions. It is very plausible that this observation
generally holds in other statistical-mechanical systems that have suitable
graphical representations.

##  VI Acknowledgments

We thank R. M. Ziff and T. M. Garoni for helpful suggestions. This research
was supported in part by NSFC under Grant No. 91024026, 11275185 and 11147013,
and the Chinese Academy of Science. We also acknowledge the Specialized
Research Fund for the Doctoral Program of Higher Education under Grant No.
20103402110053. The simulations were carried out on the NYU-ITS cluster, which
is partly supported by NSF Grant No. PHY-0424082.

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