Percolation critical exponents

In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.

Percolating systems have a parameter ${\displaystyle p\,\!}$ which controls the occupancy of sites or bonds in the system. At a critical value ${\displaystyle p_{c}\,\!}$, the mean cluster size goes to infinity and the percolation transition takes place. As one approaches ${\displaystyle p_{c}\,\!}$, various quantities either diverge or go to a constant value by a power law in ${\displaystyle |p-p_{c}|\,\!}$, and the exponent of that power law is the critical exponent. While the exponent of that power law is generally the same on both sides of the threshold, the coefficient or "amplitude" is generally different, leading to a universal amplitude ratio.

Description

Thermodynamic or configurational systems near a critical point or a continuous phase transition become fractal, and the behavior of many quantities in such circumstances is described by universal critical exponents. Percolation theory is a particularly simple and fundamental model in statistical mechanics which has a critical point, and a great deal of work has been done in finding its critical exponents, both theoretically (limited to two dimensions) and numerically.

Critical exponents exist for a variety of observables, but most of them are linked to each other by exponent (or scaling) relations. Only a few of them are independent, and the choice of the fundamental exponents depends on the focus of the study at hand. One choice is the set ${\displaystyle \{\sigma ,\,\tau \}\,\!}$ motivated by the cluster size distribution, another choice is ${\displaystyle \{d_{\text{f}},\,\nu \}\,\!}$ motivated by the structure of the infinite cluster. So-called correction exponents extend these sets, they refer to higher orders of the asymptotic expansion around the critical point.

Definitions of exponents

Self-similarity at the percolation threshold

Percolation clusters become self-similar precisely at the threshold density ${\displaystyle p_{c}\,\!}$ for sufficiently large length scales, entailing the following asymptotic power laws:

The fractal dimension ${\displaystyle d_{\text{f}}\,\!}$ relates how the mass of the incipient infinite cluster depends on the radius or another length measure, ${\displaystyle M(L)\sim L^{d_{\text{f}}}\,\!}$ at ${\displaystyle p=p_{c}\,\!}$ and for large probe sizes, ${\displaystyle L\to \infty \,\!}$. Other notation: magnetic exponent ${\displaystyle y_{t}=D=d_{f}\,\!}$ and co-dimension ${\displaystyle \Delta _{\sigma }=d-d_{f}\,\!}$.

The Fisher exponent ${\displaystyle \tau \,\!}$ characterizes the cluster-size distribution ${\displaystyle n_{s}\,\!}$, which is often determined in computer simulations. The latter counts the number of clusters with a given size (volume) ${\displaystyle s\,\!}$, normalized by the total volume (number of lattice sites). The distribution obeys a power law at the threshold, ${\displaystyle n_{s}\sim s^{-\tau }\,\!}$ asymptotically as ${\displaystyle s\to \infty \,\!}$.

The probability for two sites separated by a distance ${\displaystyle {\vec {r}}\,\!}$ to belong to the same cluster decays as ${\displaystyle g({\vec {r}})\sim |{\vec {r}}|^{-2(d-d_{\text{f}})}\,\!}$ or ${\displaystyle g({\vec {r}})\sim |{\vec {r}}|^{-d+(2-\eta )}\,\!}$ for large distances, which introduces the anomalous dimension ${\displaystyle \eta \,\!}$. Also, ${\displaystyle \delta =(d+2-\eta )/(d-2+\eta )}$ and ${\displaystyle \eta =2-\gamma /\nu }$.

The exponent ${\displaystyle \Omega \,\!}$ is connected with the leading correction to scaling, which appears, e.g., in the asymptotic expansion of the cluster-size distribution, ${\displaystyle n_{s}\sim s^{-\tau }(1+{\text{const}}\times s^{-\Omega })\,\!}$ for ${\displaystyle s\to \infty \,\!}$. Also, ${\displaystyle \omega =\Omega /(\sigma \nu )=\Omega d_{f}}$.

For quantities like the mean cluster size ${\displaystyle S\sim a_{0}|p-p_{c}|^{-\gamma }(1+a_{1}(p-p_{c})^{\Delta _{1}}+\ldots )}$, the corrections are controlled by the exponent ${\displaystyle \Delta _{1}=\Omega \beta \delta =\omega \nu }$.[1]

The minimum or chemical distance or shortest-path exponent ${\displaystyle d_{\mathrm {min} }}$ describes how the average minimum distance ${\displaystyle \langle \ell \rangle }$ relates to the Euclidean distance ${\displaystyle r}$, namely ${\displaystyle \langle \ell \rangle \sim r^{d_{\mathrm {min} }}}$ Note, it is more appropriate and practical to measure average ${\displaystyle r}$, <${\displaystyle r}$> for a given ${\displaystyle \ell }$. The elastic backbone [2] has the same fractal dimension as the shortest path. A related quantity is the spreading dimension ${\displaystyle d_{\ell }}$, which describes the scaling of the mass M of a critical cluster within a chemical distance ${\displaystyle \ell }$ as ${\displaystyle M\sim \ell ^{d_{\ell }}}$, and is related to the fractal dimension ${\displaystyle d_{f}}$ of the cluster by ${\displaystyle d_{\ell }=d_{f}/d_{\mathrm {min} }}$. The chemical distance can also be thought of as a time in an epidemic growth process, and one also defines ${\displaystyle \nu _{t}}$ where ${\displaystyle d_{\mathrm {min} }=\nu _{t}/\nu =z}$, and ${\displaystyle z}$ is the dynamical exponent.[3] One also writes ${\displaystyle \nu _{\parallel }=\nu _{t}}$.

Also related to the minimum dimension is the simultaneous growth of two nearby clusters. The probability that the two clusters coalesce exactly in time ${\displaystyle t}$ scales as ${\displaystyle p(t)\sim t^{-\lambda }}$[4] with ${\displaystyle \lambda =1+5/(4d_{\mathrm {min} })}$.[5]

The dimension of the backbone, which is defined as the subset of cluster sites carrying the current when a voltage difference is applied between two sites far apart, is ${\displaystyle d_{\text{b}}}$ (or ${\displaystyle d_{\text{BB}}}$).

The fractal dimension of the random walk on an infinite incipient percolation cluster is given by ${\displaystyle d_{w}}$.

The spectral dimension ${\displaystyle {\tilde {d}}}$ such that the average number of distinct sites visited in an ${\displaystyle N}$-step random walk scales as ${\displaystyle N^{\tilde {d}}}$.

Critical behavior close to the percolation threshold

The approach to the percolation threshold is governed by power laws again, which hold asymptotically close to ${\displaystyle p_{c}\,\!}$:

The exponent ${\displaystyle \nu \,\!}$ describes the divergence of the correlation length ${\displaystyle \xi \,\!}$ as the percolation transition is approached, ${\displaystyle \xi \sim |p-p_{c}|^{-\nu }\,\!}$. The infinite cluster becomes homogeneous at length scales beyond the correlation length; further, it is a measure for the linear extent of the largest finite cluster. Other notation: Thermal exponent ${\displaystyle y_{t}=1/\nu }$ and dimension ${\displaystyle \Delta _{\epsilon }=d-1/\nu }$.

Off criticality, only finite clusters exist up to a largest cluster size ${\displaystyle s_{\max }\,\!}$, and the cluster-size distribution is smoothly cut off by a rapidly decaying function, ${\displaystyle n_{s}\sim s^{-\tau }f(s/s_{\max })\,\!}$. The exponent ${\displaystyle \sigma }$ characterizes the divergence of the cutoff parameter, ${\displaystyle s_{\max }\sim |p-p_{c}|^{-1/\sigma }\,\!}$. From the fractal relation we have ${\displaystyle s_{\max }\sim \xi ^{d_{\text{f}}}\,\!}$, yielding ${\displaystyle \sigma =1/\nu d_{\text{f}}\,\!}$.

The density of clusters (number of clusters per site) ${\displaystyle n_{c}}$ is continuous at the threshold but its third derivative goes to infinity as determined by the exponent ${\displaystyle \alpha }$: ${\displaystyle n_{c}\sim A+B(p-p_{c})+C(p-p_{c})^{2}+D_{\pm }|p-p_{c}|^{2-\alpha }+\cdots }$, where ${\displaystyle D_{\pm }}$ represents the coefficient above and below the transition point.

The strength or weight of the percolating cluster, ${\displaystyle P}$ or ${\displaystyle P_{\infty }}$, is the probability that a site belongs to an infinite cluster. ${\displaystyle P}$ is zero below the transition and is non-analytic. Just above the transition, ${\displaystyle P\sim |p-p_{c}|^{\beta }\,\!}$, defining the exponent ${\displaystyle \beta \,\!}$. ${\displaystyle \ P}$ plays the role of an order parameter.

The divergence of the mean cluster size ${\displaystyle S=\sum _{s}s^{2}n_{s}/p_{c}\sim |p-p_{c}|^{-\gamma }\,\!}$ introduces the exponent ${\displaystyle \gamma \,\!}$.

The gap exponent Δ is defined as Δ = 1/(β+γ) = 1/σ and represents the "gap" in critical exponent values from one moment ${\displaystyle M_{n}}$ to the next ${\displaystyle M_{n+1}}$ for ${\displaystyle n>2}$.

The conductivity exponent ${\displaystyle t=t'/\nu }$ describes how the electrical conductivity ${\displaystyle C}$ goes to zero in a conductor-insulator mixture, ${\displaystyle C\sim |p-p_{c}|^{t}\,\!}$. Also, ${\displaystyle t'=\zeta }$

Surface critical exponents

The probability a point at a surface belongs to the percolating or infinite cluster is ${\displaystyle P_{\mathrm {surf} }\sim |p-p_{c}|^{\beta _{\mathrm {surf} }}\,\!}$

The surface fractal dimension is given by ${\displaystyle d_{\mathrm {surf} }=d-1-\beta _{\mathrm {surf} }/\nu }$ [6]

Scaling relations

Hyperscaling relations

${\displaystyle \tau ={\frac {d}{d_{\text{f}}}}+1\,\!}$

${\displaystyle d_{\text{f}}=d-{\frac {\beta }{\nu }}\,\!}$

${\displaystyle \eta =2+d-2d_{\text{f}}\,\!}$

Relations based on ${\displaystyle \{\sigma ,\tau \}}$

${\displaystyle \alpha =2-{\frac {\tau -1}{\sigma }}\,\!}$

${\displaystyle \beta ={\frac {\tau -2}{\sigma }}\,\!}$

${\displaystyle \gamma ={\frac {3-\tau }{\sigma }}\,\!}$

${\displaystyle \beta +\gamma ={\frac {1}{\sigma }}=\nu d_{f}\,\!}$

${\displaystyle \nu ={\frac {\tau -1}{\sigma d}}={\frac {2\beta +\gamma }{d}}\,\!}$

${\displaystyle \delta ={\frac {1}{\tau -2}}\,\!}$

Relations based on ${\displaystyle \{d_{\text{f}},\nu \}}$

${\displaystyle \alpha =2-\nu d\,\!}$

${\displaystyle \beta =\nu (d-d_{\text{f}})\,\!}$

${\displaystyle \gamma =\nu (2d_{\text{f}}-d)\,\!}$

${\displaystyle \sigma ={\frac {1}{\nu d_{\text{f}}}}\,\!}$

Conductivity scaling relations

${\displaystyle d_{w}=2+{\frac {t-\beta }{\nu }}\,\!}$

${\displaystyle t'=d_{w}-d_{f}\,\!}$

${\displaystyle {\tilde {d}}=2d_{f}/d_{w}}$

Exponents for standard percolation

d	1[7]	2	3	4	5	6 – ε [8][9][10]	6 +
α	1	–2/3	-0.625(3)	-0.756(40)		${\displaystyle -1+{\frac {\varepsilon }{7}}-{\frac {443}{2^{2}3^{2}7^{3}}}\varepsilon ^{2}}$	-1
β	0	0.14(3) [11] 5/36	0.39(2)[12] 0.4181(8), 0.41(1) [13] 0.405(25)[14] 0.4273[15]	0.52(3)[12] 0.639(20)[14] 0.657(9) 0.6590[15]	0.66(5)[12] 0.835(5)[14] 0.830(10) 0.8457[15]	${\displaystyle 1-{\frac {\varepsilon }{7}}-{\frac {61}{2^{2}3^{2}7^{3}}}\varepsilon ^{2}}$	1
γ	1	43/18	1.6[13] 1.80(5) [12] 1.66(7) [16] 1.793(3) 1.805(20) [17] 1.8357[15]	1.6(1) [12] 1.48(8)[16] 1.422(16) 1.4500[15] 1.435(15) [17]	1.3(1)[12] 1.18(7)[16] 1.185(5) [17] 1.1817[15]	${\displaystyle 1+{\frac {\varepsilon }{7}}+{\frac {565}{2^{2}3^{2}7^{3}}}\varepsilon ^{2}}$	1
δ	${\displaystyle \infty }$	91/5, 18 [18]	5.29(6) [19]^* 5.3 [18]	3.9 [18] 3.198(6) [20]	3.0 [18]	${\displaystyle 2+{\frac {2}{7}}\varepsilon +{\frac {565}{2\cdot 3^{2}7^{3}}}\varepsilon ^{2}}$	2
η	1	5/24	-0.046(8)[19] -0.059(9) [21] -0.07(5)[17] -0.0470[15]	-0.12(4)[17] -0.0944(28) [20] -0.0929(9)[22] -0.0954[15]	-0.075(20)[17] -0.0565[15]	${\displaystyle -{\frac {\varepsilon }{21}}-{\frac {206}{3^{3}7^{3}}}\varepsilon ^{2}}$	0
ν	1	1.33(5) [23] 4/3	0.8(1),[13] 0.80(5),[23] 0.872(7) [17] 0.875(1)[19] 0.8765(18)[24] 0.8960[15] 0.8764(12)[25] 0.8751(11) [26] 0.8762(12)[27] 0.8774(13)[28]	0.6782(50)[17] 0.689(10)[20] 0.6920 [15] 0.693 [29] 0.6852(28) [28] 0.6845(23) [30]	0.569(5) cited in [28] 0.571(3) [14] 0.5746 [15] 0.5723(18) [28] 0.5737(33) [30]	${\displaystyle {\frac {1}{2}}+{\frac {5}{84}}\varepsilon +{\frac {589}{2^{2}3^{3}7^{3}}}\varepsilon ^{2}}$	1/2
σ	1	36/91	0.42(6) [31] 0.445(10) [19] 0.4522(8) [20] 0.45237(8)[27] 0.4419[15]	0.476(5) 0.4742[15]	0.496(4) 0.4933[15]	${\displaystyle {\frac {1}{2}}-{\frac {1}{98}}\varepsilon ^{2}}$	1/2
τ	2	187/91	2.186(2) [21] 2.1888[15] 2.189(2) [19] 2.190(2) [22] 2.189(1) [32] 2.18906(8)[20] 2.18909(5)[27] 2.1892(1)[33]	2.26[18] 2.313(3)[34] 2.3127(6)[20] 2.313(2)[22] 2.3124[15] 2.3142(5)[33]	2.33[18] 2.412(4)[34] 2.4171[15] 2.419(1)[33]	${\displaystyle {\frac {5}{2}}-{\frac {\varepsilon }{14}}+{\frac {313}{2^{3}3^{2}7^{3}}}\varepsilon ^{2}}$	5/2
${\displaystyle d_{\text{f}}}$	1	91/48	2.523(4) [19]^* 2.530(4) [21]^* 2.5230(1) [24] 2.5226(1) [35] 2.52293(10) [27]	3.05(5), 3.003 [29] 3.0472(14)[20] 3.046(7)[34] 3.046(5)[22] 3.0479 [15] 3.0437(11)[33] 3.0446(7) [30]	3.54(4) 3.528 [15] 3.524(2)[33] 3.5260(14)[30]	${\displaystyle 4-{\frac {10}{21}}\varepsilon +{\frac {103}{3^{3}7^{3}}}\varepsilon ^{2}}$	4
Ω		0.70(2) [22] 0.77(4) [36] 0.77(2) [37] 72/91 [38][39] 0.44(9) [1]	0.50(9) [17] 0.64(2) [19] 0.73(8) [21] 0.65(2) [40] 0.60(8) [22] 0.77(3) [33] 0.64(5)[24]	0.31(5) [17] 0.5(1) [22] 0.37(4) [20] 0.4008 [15]	0.27(7) [17] 0.2034[15]	${\displaystyle {\frac {\varepsilon }{4}}-{\frac {283}{2^{2}3^{2}7^{2}}}\varepsilon ^{2}}$
ω		3/2 [38]	1.26(23) [17] 1.6334[15] 1.62(13)[24] 1.61(5)[19]	0.94(15) [17] 1.2198[15] 1.13(10) [20] 1.0(2) [41]	0.96(26) [17] 0.7178[15]	${\displaystyle \varepsilon -{\frac {671}{2\cdot 3^{2}7^{2}}}\varepsilon ^{2}}$ [42][15]	0
${\displaystyle t'=t/\nu }$		0.9479 [43] 0.995(1) [44] 0.977(8)) [45] 0.9825(8) [4]	2.276(12) [46] 2.26(4) [47] 2.305(15) [48] 2.283(3) [41]				3
${\displaystyle d_{w}}$		2.8784(8) [4]
${\displaystyle d_{s}}$		4/3 [43] 1.327(1) [44] 1.3100(11) [4]	1.32(6) [49]
${\displaystyle d_{\mathrm {surf} }}$		2/3 [50][51]	1.030(4) [52] 1.0246􏰌(4) [53]	1.32(7)[54]	1.65(3) [54]	${\displaystyle 2-{\frac {5}{21}}\varepsilon }$ [54]	2 [54]
${\displaystyle d_{\text{b}}}$		1.60(5) [2] 1.64(1) [55] 1.647(4) [3] 1.6432(8) [4] 1.6434(2) [56] 1.64336(10) [57]	1.8, 1.77(7)[2] 1.855(15)[58]	1.95(5) [59] 1.9844(11) [30]	2.00(5)[59] 2.0226(27) [30]		2
${\displaystyle d_{\text{min}}}$		1.132(2)[60] 1.130(3) [61] 1.1307(4) [3] 1.1303(8)[62] 1.1306(3) [4] 1.130 77(2) [63]	1.35(5)[2] 1.34(1) [61] 1.374(6)[52] 1.3756(6) [63] 1.3756(3) [25] 1.3755(3) [27]	1.607(5) [34] 1.6042(5) [30]	1.812(6) [34] 1.8137(16) [30]		2
${\displaystyle \lambda =\mu +1}$		2.1055(10)[64] 2.1056(3)[5] 2.1045(10)[65]

For higher-order terms in the ${\displaystyle \varepsilon }$ expansions, see [15][66]

Exponents for standard percolation on a non-trivial planar lattice (Weighted planar stochastic lattice (WPSL))

WPSL	Exponents
${\displaystyle d}$	${\displaystyle 2}$
${\displaystyle \nu }$	${\displaystyle 1.635}$
${\displaystyle \beta }$	${\displaystyle 0.222}$
${\displaystyle \gamma }$	${\displaystyle 2.825}$
${\displaystyle \tau }$	${\displaystyle 2.072}$
${\displaystyle d_{f}}$	${\displaystyle 1.864}$

Note that it has been claimed that the numerical values of exponents of percolation depend only on the dimension of lattice. However, percolation on WPSL is an exception in the sense that albeit it is two dimensional yet it does not belong to the same universality where all the planar lattices belong.[67][68]

Exponents for directed percolation

Directed percolation (DP) refers to percolation in which the fluid can flow only in one direction along bonds—such as only in the downward direction on a square lattice rotated by 45 degrees. This system is referred to as "1 + 1 dimensional DP" where the two dimensions are thought of as space and time.

${\displaystyle \nu _{\perp }}$ and ${\displaystyle \nu _{\parallel }}$ are the transverse (perpendicular) and longitudinal (parallel) correlation length exponents, respectively. Also ${\displaystyle \zeta =1/z=\nu _{\perp }/\nu _{\parallel }}$.

${\displaystyle \delta }$ is the exponent corresponding to the behavior of the survival probability as a function of time: ${\displaystyle P(t)\sim t^{-\delta }}$.

The d(space)+1(time) dimensional exponents are given below.

d+1	1+1	2+1	3+1	4 – ε [69]	Mean Field
β	0.276486(8) [70] 0.276 7(3) [71]	0.5834(30) [72] 0.580(4)[71]	0.813(9) [73] 0.818(4)[71] 0.82205[69]	${\displaystyle 1-{\frac {\varepsilon }{6}}}$	1
δ,α	0.159464(6) [70] 0.15944(2)[71]	0.4505(1) [72] 0.451(3)[74] 0.4509(5) [75] 0.4510(4) [71] 0.460(6)[76]	0.732(4) [77] 0.7398(10) [71] 0.73717 [78]	${\displaystyle 1-{\frac {\varepsilon }{4}}}$	1
η,θ	0.313686(8) [70] 0.31370(5) [71]	0.2303(4) [75] 0.2307(2) [71] 0.2295(10) [72] 0.229(3) [74] 0.214(8) [76]	0.1057(3)[71] 0.114(4) [73] 0.12084 [78]
${\displaystyle \nu _{\parallel }}$	1.733847(6) [70] 1.7355(15) [71] 1.73(2)[79]	1.16(5)[79] 1.287(2) [71] 1.295(6) [74]	1.106(3) [71] 1.11(1) [73] 1.10571 [78]
${\displaystyle \nu _{\perp }}$	1.096854(4) [70] 1.0979(10) [71]	0.7333(75) [77] 0.729(1) [71]	0.584(5) [77] 0.582(2) [71] 0.58360 [78]	${\displaystyle {\frac {1}{2}}+{\frac {\varepsilon }{16}}}$	${\displaystyle {\frac {1}{2}}}$
z	1.580745(10) [70] 1.5807(2) [71]	1.1325(10) [72] 1.133(2) [74] 1.134(4) [76] 1.7660(16)[77] 1.765(3)[74] 1.766(2) [72] 1.7665(2) [71] 1.7666(10) [75]	1.88746 [78] 1.8990(4) [71] 1.901(5) [77]	${\displaystyle 2-{\frac {\varepsilon }{12}}}$	2
γ	2.277730(5) = 41/18?,[70] 2.278(2) [80]	1.595(18) [72]	1.237(23) [73]		1

Scaling relations for directed percolation

${\displaystyle \delta =\beta /\nu _{\parallel }}$

${\displaystyle \eta =\gamma /\nu _{\parallel }-1}$

${\displaystyle \Delta =\beta +\gamma }$

Exponents for percolation on networks

The percolation critical exponents of Erdos Reyni networks are the same as for mean field or as for d=6 and above. However, for scale free networks they are very different because of breaking their structural symmetry[81][82] . Nodes with low degrees have different neighborhood compared to high degree nodes. Percolation of networks composed of communities can be mapped to percolation or magnetism under external field.[83] For interdependent networks one obtains percolation of first order transition.[84] Also in k-core percolation one obtains a first order percolation transition.[85] The presence of communities, in these system can be regarded as an external field.[86]

See also

• Critical exponent
• Percolation theory
• Percolation threshold
• Conductivity near the percolation threshold

References

1. Adler, Joan; Moshe, Moshe; Privman, Vladimir (1983). "Chapter 2: Corrections to Scaling for Percolation". In Deustscher, G.; Zallen, R.; Adler, J. (eds.). Percolation Structures and Processes, Ann. Israel Phys. Soc. 5. Adam Hilger, Bristol. pp. 397–423.
2. Herrmann, H. J.; D. C. Hong; H. E. Stanley (1984). "Backbone and elastic backbone of percolation clusters obtained by the new method of 'burning'". J. Phys. A: Math. Gen. 17 (5): L261–L266. Bibcode:1984JPhA...17L.261H. doi:10.1088/0305-4470/17/5/008. S2CID 16510317.
3. Grassberger, Peter (1992). "Spreading and backbone dimensions of 2D percolation". J. Phys. A: Math. Gen. 25 (21): 5475–5484. Bibcode:1992JPhA...25.5475G. doi:10.1088/0305-4470/25/21/009.
4. Grassberger, Peter (1999). "Conductivity exponent and backbone dimension in 2-d percolation". Physica A. 262 (3–4): 251–263. arXiv:cond-mat/9808095. Bibcode:1999PhyA..262..251G. doi:10.1016/S0378-4371(98)00435-X. S2CID 955125.
5. Ziff, R. M. (1999). "Exact critical exponent for the shortest-path scaling function in percolation". J. Phys. A: Math. Gen. 32 (43): L457–L459. arXiv:cond-mat/9907305. Bibcode:1999JPhA...32L.457Z. doi:10.1088/0305-4470/32/43/101. S2CID 1605985.
6. Stauffer, D.; A. Aharony (1999). "Density profile of the incipient infinite percolation cluster". International Journal of Modern Physics C. 10 (5): 935–940. Bibcode:1999IJMPC..10..935S. doi:10.1142/S0129183199000735.
7. Reynolds, P. J.; H. E. Stanley; W. Klein (1977). "Ghost fields, pair connectedness, and scaling: exact results in one-dimensional percolation". Journal of Physics A: Mathematical and General. 10 (11): L203–L209. doi:10.1088/0305-4470/10/11/007.
8. Essam, J. W. (1980). "Percolation theory". Rep. Prog. Phys. 43 (7): 833–912. Bibcode:1980RPPh...43..833E. doi:10.1088/0034-4885/43/7/001.
9. Harris, A. B.; T. C. Lubensky; W. K. Holcomb; C. Dasgupta (1975). "Renormalization-group approach to percolation problems". Physical Review Letters. 35 (6): 327–330. Bibcode:1975PhRvL..35..327H. doi:10.1103/PhysRevLett.35.327.
10. Harris, A. B.; T. C. Lubensky; W. K. Holcomb; C. Dasgupta (1975). "Renormalization-group approach to percolation problems". Physical Review Letters. 35 (6): 327–330. Bibcode:1975PhRvL..35..327H. doi:10.1103/PhysRevLett.35.327.
11. Sykes, M. F.; M. Glen; D. S. Gaunt (1974). "The percolation probability for the site problem on the triangular lattice". J. Phys. A: Math. Gen. 7 (9): L105–L108. Bibcode:1974JPhA....7L.105S. doi:10.1088/0305-4470/7/9/002.
12. Kirkpatrick, Scott (1976). "Percolation phenomena in higher dimensions: Approach to the mean-field limit". Phys. Rev. Lett. 36 (2): 69–72. doi:10.1103/PhysRevLett.36.69.
13. Sur, A.; Joel L. Lebowitz; J. Marro; M. H. Kalos; S. Kirkpatrick (1976). "Monte Carlo Studies of Percolation Phenomena for a Simple Cubic Lattice". J. Stat. Phys. 15 (5): 345–353. Bibcode:1976JSP....15..345S. doi:10.1007/BF01020338. S2CID 38734613.
14. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris; Lior Klein (1990). "Low-Concentration Series in General Dimension". Journal of Statistical Physics. 58 (3/4): 511–538. Bibcode:1990JSP....58..511A. doi:10.1007/BF01112760. S2CID 122109020.
15. Gracey, J. A. (2015). "Four loop renormalization of φ^3 theory in six dimensions". Phys. Rev. D. 92 (2): 025012. arXiv:1506.03357. Bibcode:2015PhRvD..92b5012G. doi:10.1103/PhysRevD.92.025012. S2CID 119205590.
16. Gaunt, D. S.; H. Ruskin (1978). "Bond percolation processes in d dimensions". J. Phys. A: Math. Gen. 11 (7): 1369–1380. doi:10.1088/0305-4470/11/7/025.
17. Adler, J.; Y. Meir; A. Aharony; A.B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Phys. Rev. E. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID 9993262.
18. Nakanishi, H; H. E. Stanley (1980). "Scaling studies of percolation phenomena in systems of dimensionality of two to seven: Cluster numbers". Physical Review B. 22 (5): 2466–2488. Bibcode:1980PhRvB..22.2466N. doi:10.1103/PhysRevB.22.2466.
19. Lorenz, C. D.; R. M. Ziff (1998). "Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices". Phys. Rev. E. 57 (1): 230–236. arXiv:cond-mat/9710044. Bibcode:1998PhRvE..57..230L. doi:10.1103/PhysRevE.57.230. S2CID 119074750.
20. Ballesteros, H. G.; L. A. Fernández; V. Martín-Mayor; A. Muñoz Sudepe; G. Parisi; J. J. Ruiz-Lorenzo (1997). "Measures of critical exponents in the four-dimensional site percolation". Physics Letters B. 400 (3–4): 346–351. arXiv:hep-lat/9612024. Bibcode:1997PhLB..400..346B. doi:10.1016/S0370-2693(97)00337-7. S2CID 10242417.
21. Jan, N.; D. Stauffer (1998). "Random Site Percolation in Three Dimensions". Int. J. Mod. Phys. C. 9 (2): 341–347. Bibcode:1998IJMPC...9..341J. doi:10.1142/S0129183198000261.
22. Tiggemann, D. (2001). "Simulation of percolation on massively parallel computers". Int. J. Mod. Phys. C. 12 (6): 871–878. arXiv:cond-mat/0106354. Bibcode:2001IJMPC..12..871T. doi:10.1142/S012918310100205X. S2CID 118911971.
23. Levenshteĭn, M. E.; B. I. Shklovskiĭ; M. S. Shur; A. L. Éfros (1975). "The relation between the critical exponents of percolation theory". Zh. Eksp. Teor. Fiz. 69: 386–392. Bibcode:1976JETP...42..197L.,
24. Ballesteros, P. N.; L. A. Fernández; V. Martín-Mayor; A. Muñoz Sudepe; G. Parisi; J. J. Ruiz-Lorenzo (1999). "Scaling corrections: site percolation and Ising model in three dimensions". Journal of Physics A. 32 (1): 1–13. arXiv:cond-mat/9805125. Bibcode:1999JPhA...32....1B. doi:10.1088/0305-4470/32/1/004. S2CID 2787294.
25. Wang, J.; Z. Zhou; W. Zhang; T. M. Garoni; Y. Deng (2013). "Bond and site percolation in three dimensions". Physical Review E. 87 (5): 052107. arXiv:1302.0421. Bibcode:2013PhRvE..87e2107W. doi:10.1103/PhysRevE.87.052107. PMID 23767487. S2CID 14087496.,
26. Hu, H.; H. W. Blöte; R. M. Ziff; Y. Deng (2014). "Short-range correlations in percolation at criticality". Physical Review E. 90 (4): 042106. arXiv:1406.0130. Bibcode:2014PhRvE..90d2106H. doi:10.1103/PhysRevE.90.042106. PMID 25375437. S2CID 21410490.
27. Xu, Xiao; Wang, Junfeng; Lv, Jian-Ping; Deng, Youjin (2014). "Simultaneous analysis of three-dimensional percolation models". Frontiers of Physics. 9 (1): 113–119. arXiv:1310.5399. Bibcode:2014FrPhy...9..113X. doi:10.1007/s11467-013-0403-z. S2CID 119250232.
28. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056.
29. LeClair, André; Joshua Squires (2018). "Conformal bootstrap for percolation and polymers". Journal of Statistical Mechanics: Theory and Experiment. 2018 (12): 123105. arXiv:1802.08911. Bibcode:2018arXiv180208911L. doi:10.1088/1742-5468/aaf10a. S2CID 73674896.
30. Zhang, Zhongjin; Pengcheng Hou; Sheng Fang; Hao Hu; Youjin Deng (2020). "Critical exponents and universal excess cluster number of percolation in four and five dimensions". Preprint. arXiv:2004.11289.
31. Sykes, M. F.; D. S. Gaunt; J. W. Essam (1976). "The percolation probability for the site problem on the face-centred cubic lattice". J. Phys. A: Math. Gen. 9 (5): L43–L46. Bibcode:1976JPhA....9L..43S. doi:10.1088/0305-4470/9/5/002.
32. Tiggemann, D. (2006). "Percolation on growing lattices". Int. J. Mod. Phys. C. 17 (8): 1141–1150. arXiv:cond-mat/0604418. Bibcode:2006IJMPC..17.1141T. doi:10.1142/S012918310600962X. S2CID 119398198.
33. Mertens, Stephan; Cristopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851.
34. Paul, Gerald; R. M. Ziff; H. E. Stanley (2001). "Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions". Phys. Rev. E. 64 (2): 026115. arXiv:cond-mat/0101136. Bibcode:2001PhRvE..64b6115P. doi:10.1103/PhysRevE.64.026115. PMID 11497659. S2CID 18271196.
35. Deng, Youjin; Henk W. J. Blöte (2005). "Monte Carlo study of the site-percolation model in two and three dimensions". Phys. Rev. E. 72 (1): 016126. Bibcode:2005PhRvE..72a6126D. doi:10.1103/PhysRevE.72.016126. PMID 16090055.
36. Kammerer, A.; F. Höfling; T. Franosch (2008). "Cluster-resolved dynamic scaling theory and universal corrections for transport on percolating systems". Europhys. Lett. 84 (6): 66002. arXiv:0811.1414. Bibcode:2008EL.....8466002K. doi:10.1209/0295-5075/84/66002. S2CID 16581770.
37. Ziff, R. M.; F. Babalievski (1999). "Site percolation on the Penrose rhomb lattice". Physica A. 269 (2–4): 201–210. Bibcode:1999PhyA..269..201Z. doi:10.1016/S0378-4371(99)00166-1.
38. Ziff, R. M. (2011). "Correction-to-scaling exponent for two-dimensional percolation". Phys. Rev. E. 83 (2): 020107. arXiv:1101.0807. Bibcode:2011PhRvE..83b0107Z. doi:10.1103/PhysRevE.83.020107. PMID 21405805. S2CID 14750620.
39. Aharony, Amnon; Asikainen, Joonas (2003). "Fractal dimension and corrections to scaling for critical Potts clusters". Fractals, Supplementary Issue. 11 (1): 3–7. arXiv:cond-mat/0206367. doi:10.1142/S0218348X03001665.
40. Gimel, Jean-Christophe; Taco Nicolai; Dominique Durand (2000). "Size distribution of percolating clusters on cubic lattices". J. Phys. A: Math. Gen. 33 (43): 7687–7697. Bibcode:2000JPhA...33.7687G. doi:10.1088/0305-4470/33/43/302.
41. Kozlov, B.; M. Laguës (2010). "Universality of 3D percolation exponents and first-order corrections to scaling for conductivity exponents". Physica A. 389 (23): 5339–5346. Bibcode:2010PhyA..389.5339K. doi:10.1016/j.physa.2010.08.002.
42. Houghton, A.; J. S. Reeve; D. J. Wallace (1978). "High-order behavior in phi^3 field theories and the percolation problem". Phys. Rev. B. 17 (7): 2956. Bibcode:1978PhRvB..17.2956H. doi:10.1103/PhysRevB.17.2956.
43. Alexander, S.; R. Orbach (1982). "Density of states on fractals : 'fractons'" (PDF). Journal de Physique Lettres. 43 (17): L625–L631. doi:10.1051/jphyslet:019820043017062500.
44. Milovanov, A. V. (1997). "Topological proof for the Alexander-Orbach conjecture". Phys. Rev. E. 56 (3): 2437–2446. Bibcode:1997PhRvE..56.2437M. doi:10.1103/PhysRevE.56.2437.
45. Cen, Wei; Dongbing Liu; Bingquan Mao (2012). "Molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions". Physica A. 391 (4): 925–929. Bibcode:2012PhyA..391..925C. doi:10.1016/j.physa.2011.01.003.
46. Gingold, David B.; C. J. Lobb (1990). "Percolative conduction in three dimensions". Physical Review B. 42 (13): 8220. Bibcode:1990PhRvB..42.8220G. doi:10.1103/PhysRevB.42.8220. PMID 9994994.
47. Normand, Jean-Marie; Hans J. Herrmann (1995). "Precise determination of the conductivity exponent of 3D percolation using "Percola"". International Journal of Modern Physics C. 6 (6): 813. arXiv:cond-mat/9602081. Bibcode:1995IJMPC...6..813N. doi:10.1142/S0129183195000678. S2CID 2912863.
48. Clerc, Jean-Marie; V. A. Podolskiy; A. K. Sarychev (2000). "Precise determination of the conductivity exponent of 3D percolation using exact numerical renormalization". The European Physical Journal B. 15 (3): 507–516. Bibcode:2000EPJB...15..507C. doi:10.1007/s100510051153. S2CID 121306901.
49. Argyrakis, P.; R. Kopelman (1984). "Random walk on percolation clusters". Physical Review B. 29 (1): 511–514. Bibcode:1984PhRvB..29..511A. doi:10.1103/PhysRevB.29.511.
50. Cardy, John (1984). "Conformal invariance and surface critical behavior". Nuclear Physics B. 240 (4): 514–532. Bibcode:1984NuPhB.240..514C. doi:10.1016/0550-3213(84)90241-4.
51. Vanderzande, C. (1988). "Surface fractal dimension of two-dimensional percolation". J. Phys. A: Math. Gen. 21 (3): 833–837. Bibcode:1988JPhA...21..833V. doi:10.1088/0305-4470/21/3/039.
52. Grassberger, Peter (1992). "Numerical studies of critical percolation in three dimensions". J. Phys. A: Math. Gen. 25 (22): 5867–5888. Bibcode:1992JPhA...25.5867G. doi:10.1088/0305-4470/25/22/015.
53. Deng, Youjin; Henk W. J. Blöte (2005). "Surface critical phenomena in three-dimensional percolation". Phys. Rev. E. 71 (1): 016117. Bibcode:2005PhRvE..71a6117D. doi:10.1103/PhysRevE.71.016117. PMID 15697668.
54. Diehl, H. W.; P. M.Lam (1989). "Semi-infinite Potts model and percolation at surfaces". Z. Phys. B. 74 (3): 395–401. Bibcode:1989ZPhyB..74..395D. doi:10.1007/BF01307889. S2CID 121559161.
55. Rintoul, M. D.; H. Nakanishi (1992). "A precise determination of the backbone fractal dimension on two-dimensional percolation clusters". J. Phys. A: Math. Gen. 25 (15): L945. doi:10.1088/0305-4470/25/15/008.
56. Deng, Youjin; Henk W. J. Blöte; Bernard Neinhuis (2004). "Backbone exponents of the two-dimensional q-state Potts model: A Monte Carlo investigation". Phys. Rev. E. 69 (2): 026114. Bibcode:2004PhRvE..69b6114D. doi:10.1103/PhysRevE.69.026114. PMID 14995527.
57. Xu, Xiao; Wang, Junfeng; Zhou, Zongzheng; Garoni, Timothy M.; Deng, Youjin (2014). "Geometric structure of percolation clusters". Physical Review E. 89 (1): 012120. arXiv:1309.7244. Bibcode:2014PhRvE..89a2120X. doi:10.1103/PhysRevE.89.012120. PMID 24580185. S2CID 25468743.
58. Rintoul, M. D.; H. Nakanishi (1994). "A precise characterization of three-dimensional percolating backbones". J. Phys. A: Math. Gen. 27 (16): 5445–5454. Bibcode:1994JPhA...27.5445R. doi:10.1088/0305-4470/27/16/011.
59. Moukarzel, C. (1994). "A Fast Algorithm for Backbones". Int. J. Mod. Phys. C. 9 (6): 887–895. arXiv:cond-mat/9801102. doi:10.1142/S0129183198000844. S2CID 14077176.
60. Grassberger, P. (1985). "On the spreading of two-dimensional percolation". J. Phys. A: Math. Gen. 18 (4): L215–L219. Bibcode:1985JPhA...18L.215G. doi:10.1088/0305-4470/18/4/005.
61. Herrmann, Hans J.; H. Eugene Stanley (1988). "The fractal dimension of the minimum path in two- and three-dimensional percolation". J. Phys. A: Math. Gen. 21: L829–L833. Bibcode:1984JPhA...17L.261H. doi:10.1088/0305-4470/17/5/008. S2CID 16510317.
62. Deng, Youjin; Wei Zhang; Timothy M. Garoni; Alan D. Sokal; Andrea Sportiello (2010). "Some geometric critical exponents for percolation and the random-cluster model". Physical Review E. 81 (2): 020102(R). arXiv:0904.3448. Bibcode:2010PhRvE..81b0102D. doi:10.1103/PhysRevE.81.020102. PMID 20365513. S2CID 1746746.
63. Zhou, Zongzheng; Ji Yang; Youjin Deng; Robert M. Ziff (2012). "Shortest-path fractal dimension for percolation in two and three dimensions". Physical Review E. 86 (6): 061101. arXiv:1110.1955. Bibcode:2012PhRvE..86a1101G. doi:10.1103/PhysRevE.86.061101. PMID 23367887. S2CID 37986944.
64. Grassberger, Peter (1999). "Pair connectedness and shortest-path scaling in critical percolation". J. Phys. A: Math. Gen. 32 (35): 6233–6238. arXiv:cond-mat/9906309. Bibcode:1999JPhA...32.6233G. doi:10.1088/0305-4470/32/35/301. S2CID 17663911.
65. Brereton, Tim; Christian Hirsch; Volker Schmidt; Dirk Kroese (2014). "A critical exponent for shortest-path scaling in continuum percolation". J. Phys. A: Math. Theor. 47 (50): 505003. Bibcode:2014JPhA...47X5003B. doi:10.1088/1751-8113/47/50/505003. S2CID 14191555.
66. Alcantara Bonfim, 0. F.; J E Kirkham; A J McKane (1981). "Critical exponents for the percolation problem and the Yang-Lee edge singularity". J. Phys. A: Math. Gen. 14 (9): 2391–2413. Bibcode:1981JPhA...14.2391D. doi:10.1088/0305-4470/14/9/034.
67. Hassan, M. K.; Rahman, M. M. (2015). "Percolation on a multifractal scale-free planar stochastic lattice and its universality class". Physical Review E. 92 (4): 040101. arXiv:1504.06389. Bibcode:2015PhRvE..92d0101H. doi:10.1103/PhysRevE.92.040101. PMID 26565145. S2CID 119112286.
68. Hassan, M. K.; Rahman, M. M. (2016). "Universality class of site and bond percolation on multifractal scale-free planar stochastic lattice". Physical Review E. 94 (4): 042109. arXiv:1604.08699. Bibcode:2016PhRvE..94d2109H. doi:10.1103/PhysRevE.94.042109. PMID 27841467. S2CID 22593028.
69. Janssen, H. K.; Täuber, U. C. (2005). "The field theory approach to percolation processes". Annals of Physics. 315 (1): 147–192. arXiv:cond-mat/0409670. Bibcode:2005AnPhy.315..147J. doi:10.1016/j.aop.2004.09.011. S2CID 19033621.
70. Jensen, I. (1999). "Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice". J. Phys. A. 32 (48): 5233–5249. arXiv:cond-mat/9906036. Bibcode:1999JPhA...32.5233J. doi:10.1088/0305-4470/32/28/304. S2CID 2681356.
71. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "High-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Phys. Rev. E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467.
72. Voigt, C. A.; Ziff, R. M. (1997). "Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model". Phys. Rev. E. 56 (6): R6241–R6244. arXiv:cond-mat/9710211. Bibcode:1997PhRvE..56.6241V. doi:10.1103/PhysRevE.56.R6241. S2CID 118952705.
73. Jensen, I. (1992). "Critical behavior of the three-dimensional contact process". Phys. Rev. A. 45 (2): R563–R566. Bibcode:1992PhRvA..45..563J. doi:10.1103/PhysRevA.45.R563. PMID 9907104.
74. Grassberger, P.; Y. Zhang (1996). "'Self-organized' formulation of standard percolation phenomena". Physica A. 224 (1–2): 169. Bibcode:1996PhyA..224..169G. doi:10.1016/0378-4371(95)00321-5.
75. Perlsman, E.; S. Havlin (2002). "Method to estimate critical exponents using numerical studies". EPL. 58 (2): 176–181. Bibcode:2002EL.....58..176P. doi:10.1209/epl/i2002-00621-7. S2CID 67818664.
76. Grassberger, P. (1989). "Directed percolation in 2+1 dimensions". J. Phys. A: Math. Gen. 22 (17): 3673–3679. Bibcode:1989JPhA...22.3673G. doi:10.1088/0305-4470/22/17/032.
77. Henkel, M.; H. Hinrichsen; S. Lŭbeck (2008). Non-equilibrium phase transitions, Vol. 1: Absorbing phase transitions. Springer, Dordrecht.
78. Janssen, H. K. (1981). "On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state". Annals of Physics. 42 (2): 151–154. Bibcode:1981ZPhyB..42..151J. doi:10.1007/BF01319549. S2CID 120819248.
79. Amaral, L. A. N.; A.-L. Barabási; S. V. Buldyrev; S. T. Harrington; S. Havlin; R. Sadr-Lahijany; H. E. Stanley (1995). "Avalanches and the directed percolation depinning model: Experiments, simulations, and theory". Phys. Rev. E. 51 (5): 4655–4673. arXiv:cond-mat/9412047. Bibcode:1995PhRvE..51.4655A. doi:10.1103/PhysRevE.51.4655. PMID 9963178. S2CID 9953616.
80. Essam, J. W.; A. J. Guttmann; K. De'Bell (1988). "On two-dimensional directed percolation". J. Phys. A. 21 (19): 3815–3832. Bibcode:1988JPhA...21.3815E. doi:10.1088/0305-4470/21/19/018.
81. Cohen, R; Ben-Avraham, D; Havlin, S (2002). "Percolation critical exponents in scale-free networks". Physical Review E. 66 (3): 036113. arXiv:cond-mat/0202259. doi:10.1103/PhysRevE.66.036113. PMID 12366190. S2CID 678598.
82. Cohen, Reuven; Havlin, Shlomo (2010). "Complex Networks: Structure, Robustness and Function". Cambridge Univ. Press. doi:10.1017/CBO9780511780356. ISBN 9780511780356.
83. Dong, Gaogao; Fan, Jingfang; Shekhtman, Louis M; Shai, Saray; Du, uijin; Tian, Lixin; Chen, Xiaosong; Stanley, H Eugene; Havlin, Shlomo (2018). "Resilience of networks with community structure behaves as if under an external field". Proceedings of the National Academy of Sciences. 115 (27): 6911–6915. arXiv:1805.01032. doi:10.1073/pnas.1801588115. PMC 6142202. PMID 29925594.
84. S.V. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin (2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464: 08932.CS1 maint: multiple names: authors list (link)
85. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes (2006). ", K-Core Organization of Complex Networks,". Phys. Rev. Lett. 96: 040601.CS1 maint: multiple names: authors list (link)
86. Bnaya Gross, Hillel Sanhedrai, Louis Shekhtman, Shlomo Havlin (2020). "Interconnections between networks acting like an external field in a first-order percolation transition". Physical Review E. 101 (2): 022316.CS1 maint: multiple names: authors list (link)

Further reading

• Stauffer, D.; Aharony, A. (1994), Introduction to Percolation Theory (2nd ed.), CRC Press, ISBN 978-0-7484-0253-3
• ben-Avraham, D.; Havlin, S. (2000), Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, ISBN 978-0-521-61720-8
• Bunde, A.; Havlin, S. (1996), Fractals and Disordered Systems, Springer, ISBN 978-3-642-84868-1