Percolation critical exponents

For more information, see Percolation theory .

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 ${\displaystyle y_{h}=D=d_{f}\,\!}$ 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 ${\displaystyle d_{\text{BB}}}$). One also defines ${\displaystyle \xi =d-d_{\text{b}}}$. ^[6]

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 ${\displaystyle t=\nu t'}$ 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 for ${\displaystyle p\geq p_{c}}$ is ${\displaystyle P_{\mathrm {surf} }\sim (p-p_{c})^{\beta _{\mathrm {surf} }}\,\!}$.

The surface fractal dimension is given by ${\displaystyle d_{\mathrm {surf} }=d-1-\beta _{\mathrm {surf} }/\nu }$. ^[7]

Correlations parallel and perpendicular to the surface decay as ${\displaystyle g_{\parallel }({\vec {r}})\sim |{\vec {r}}|^{2-d-\eta _{\parallel }}\,\!}$ and ${\displaystyle g_{\perp }({\vec {r}})\sim |{\vec {r}}|^{2-d-\eta _{\perp }}\,\!}$. ^[8]

The mean size of finite clusters connected to a site in the surface is ${\displaystyle \chi _{1}\sim |p-p_{c}|^{-\gamma _{1}}}$. ^{[9] [10] [11]}

The mean number of surface sites connected to a site in the surface is ${\displaystyle \chi _{1,1}\sim |p-p_{c}|^{-\gamma _{1,1}}}$. ^{[9] [10] [11]}

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

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

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

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

Surface scaling relations

${\displaystyle \eta _{\parallel }=2-d+2\beta _{\mathrm {surf} }/\nu \,\!}$

${\displaystyle d_{\mathrm {surf} }=d-1-\beta _{\mathrm {surf} }/\nu \,\!}$

${\displaystyle \eta _{\parallel }=2\eta _{\perp }-\eta \,}$ ^[12]

${\displaystyle \gamma _{1}=\nu (2-\eta _{\perp })\,\!}$ ^[11]

${\displaystyle \gamma _{1,1}=\nu (d-1-2\beta _{\mathrm {surf} }/\nu )=\nu (1-\eta _{\parallel })\,\!}$ ^{[11] [13]}

${\displaystyle \gamma +\nu =2\gamma _{1}-\gamma _{1,1}\,\!}$ ^{[10] [11]}

${\displaystyle x_{1}=\beta _{\mathrm {surf} }/\nu \,\!}$

Exponents for standard percolation

d	1 [14]	2	3	4	5	6 – ε [15] [16] [17] [note 1]	6 +
α	1	–2/3	-0.625(3) -0.64(4) [20]	-0.756(40) -0.75(2) [20]	-0.870(1) [20]	${\displaystyle -1+{\frac {\varepsilon }{7}}-{\frac {443}{2^{2}3^{2}7^{3}}}\varepsilon ^{2}}$	-1
β	0	0.14(3) [21] 5/36	0.39(2) [22] 0.4181(8) 0.41(1) [23] 0.405(25), [24] 0.4273 [19] 0.4053(5) [25] 0.429(4) [20]	0.52(3) [22] 0.639(20) [26] 0.657(9) 0.6590 [19] 0.658(1) [20]	0.66(5) [22] 0.835(5) [26] 0.830(10) 0.8457 [19] 0.8454(2) [20]	${\displaystyle 1-{\frac {\varepsilon }{7}}-{\frac {61}{2^{2}3^{2}7^{3}}}\varepsilon ^{2}}$	1
γ	1	43/18	1.6 [23] 1.80(5) [22] 1.66(7) [27] 1.793(3) 1.805(20) [26] 1.8357 [19] 1.819(3) [25] 1.78(3) [20]	1.6(1) [22] 1.48(8) [27] 1.422(16) 1.4500 [19] 1.435(15) [26] 1.430(6) [20]	1.3(1) [22] 1.18(7) [27] 1.185(5) [26] 1.1817 [19] 1.1792(7) [20]	${\displaystyle 1+{\frac {\varepsilon }{7}}+{\frac {565}{2^{2}3^{2}7^{3}}}\varepsilon ^{2}}$	1
δ	${\displaystyle \infty }$	91/5, 18 [28]	5.29(6) [29] * 5.3 [28] 5.16(4) [20]	3.9 [28] 3.198(6) [30] 3.175(8) [20]	3.0 [28] 2.3952(12) [20]	${\displaystyle 2+{\frac {2}{7}}\varepsilon +{\frac {565}{2\cdot 3^{2}7^{3}}}\varepsilon ^{2}}$	2
η	1	5/24	-0.046(8) [29] -0.059(9) [31] -0.07(5) [26] -0.0470 [19] −0.03(1) [20]	-0.12(4) [26] -0.0944(28) [30] -0.0929(9) [32] -0.0954 [19] -0.084(4) [20]	-0.075(20) [26] -0.0565 [19] −0.0547(10) [20]	${\displaystyle -{\frac {\varepsilon }{21}}-{\frac {206}{3^{3}7^{3}}}\varepsilon ^{2}}$	0
ν	1	1.33(5) [33] 4/3	0.8(1), [23] 0.80(5), [33] 0.872(7) [26] 0.875(1) [29] 0.8765(18) [34] 0.8960 [19] 0.8764(12) [35] 0.8751(11) [36] 0.8762(12) [37] 0.8774(13) [38] 0.88(2) [20]	0.6782(50) [26] 0.689(10) [30] 0.6920 [19] 0.693 [39] 0.6852(28) [38] 0.6845(23) [40] 0.6845(6) [41] 0.686(2) [20]	0.51(5) [42] 0.569(5) cited in [38] 0.571(3) [26] 0.5746 [19] 0.5723(18) [38] 0.5737(33) [40] 0.5757(7) [41] 0.5739(1) [20]	${\displaystyle {\frac {1}{2}}+{\frac {5}{84}}\varepsilon +{\frac {589}{2^{2}3^{3}7^{3}}}\varepsilon ^{2}}$	1/2
σ	1	36/91	0.42(6) [43] 0.445(10) [29] 0.4522(8) [30] 0.4524(6) [37] 0.4419 [19] 0.452(7) [20]	0.476(5) 0.4742 [19] 0.4789(14) [20]	0.496(4) 0.4933 [19] 0.49396(13) [20]	${\displaystyle {\frac {1}{2}}-{\frac {1}{98}}\varepsilon ^{2}}$	1/2
τ	2	187/91	2.186(2) [31] 2.1888 [19] 2.189(2) [29] 2.190(2) [32] 2.189(1) [44] 2.18906(8) [30] 2.18909(5) [37] 2.1892(1) [45] 2.1938(12) [20]	2.26 [28] 2.313(3) [46] 2.3127(6) [30] 2.313(2) [32] 2.3124 [19] 2.3142(5) [45] 2.3150(8) [20]	2.33 [28] 2.412(4) [46] 2.4171 [19] 2.419(1) [45] 2.4175(2) [20]	${\displaystyle {\frac {5}{2}}-{\frac {\varepsilon }{14}}+{\frac {313}{2^{3}3^{2}7^{3}}}\varepsilon ^{2}}$	5/2
${\displaystyle d_{\text{f}}}$	1	91/48	2.523(4) [29] * 2.530(4) [31] * 2.5230(1) [34] 2.5226(1) [47] 2.52293(10) [37]	3.12(2) [42] , 3.05(5), 3.003 [39] 3.0472(14) [30] 3.046(7) [46] 3.046(5) [32] 3.0479 [19] 3.0437(11) [45] 3.0446(7) [40]	3.54(4) 3.69(2) [42] 3.528 [19] 3.524(2) [45] 3.5260(14) [40]	${\displaystyle 4-{\frac {10}{21}}\varepsilon +{\frac {103}{3^{3}7^{3}}}\varepsilon ^{2}}$	4
Ω		0.70(2) [32] 0.77(4) [48] 0.77(2) [49] 72/91 [50] [51] 0.44(9) [1]	0.50(9) [26] 0.64(2) [29] 0.73(8) [31] 0.65(2) [52] 0.60(8) [32] 0.77(3) [45] 0.64(5) [34]	0.31(5) [26] 0.5(1) [32] 0.37(4) [30] 0.4008 [19]	0.27(7) [26] 0.2034 [19] 0.210(2) [20]	${\displaystyle {\frac {\varepsilon }{4}}-{\frac {283}{2^{2}3^{2}7^{2}}}\varepsilon ^{2}}$
ω		3/2 [50]	1.26(23) [26] 1.6334 [19] 1.62(13) [34] 1.61(5) [29]	0.94(15) [26] 1.2198 [19] 1.13(10) [30] 1.0(2) [53]	0.96(26) [26] 0.7178 [19]	${\displaystyle \varepsilon -{\frac {671}{2\cdot 3^{2}7^{2}}}\varepsilon ^{2}}$ [54] [19]	0
${\displaystyle t'=t/\nu }$		0.9479 [55] 0.995(1) [56] 0.977(8)) [57] 0.9825(8) [4]	2.276(12) [58] 2.26(4) [59] 2.305(15) [60] 2.283(3) [53]				3
${\displaystyle d_{w}}$		2.8784(8) [4]
${\displaystyle d_{s}}$		4/3 [55] 1.327(1) [56] 1.3100(11) [4]	1.32(6) [61]
${\displaystyle d_{\mathrm {surf} }}$		2/3 [62] [63]	1.04(5) [10] 1.030(6) [64] 1.0246(4) [65]	1.32(7) [66]	1.65(3) [66]	${\displaystyle 2-{\frac {5}{21}}\varepsilon }$ [66]	2 [66]
${\displaystyle \beta _{\mathrm {surf} }/\nu =x_{1}}$		1/3 [62]	0.98(2) [67] 0.970(6) [64] 0.975(4) [68] 0.9754(4) [65] 0.974(2) [69]	1.64(2) [69]	2.408(5) [69]	3
${\displaystyle \eta _{\parallel }}$ (surf)		2/3 [62]	1.02(12) [66] 1.08(10) [10]	1.37(13) [66]	1.7(6) [66]
${\displaystyle d_{\text{b}}}$		1.60(5) [2] 1.64(1) [70] 1.647(4) [3] 1.6432(8) [4] 1.6434(2) [71] 1.64336(10) [72] 1.64333316328711...* [6]	1.8, 1.77(7) [2] 1.855(15) [73]	1.95(5) [74] 1.9844(11) [40]	2.00(5) [74] 2.0226(27) [40]		2
${\displaystyle d_{\text{min}}=z}$		1.132(2) [75] 1.130(3) [76] 1.1307(4) [3] 1.1303(8) [77] 1.1306(3) [4] 1.130 77(2) [78]	1.35(5) [2] 1.34(1) [76] 1.374(6) [64] 1.3756(6) [78] 1.3756(3) [35] 1.3755(3) [37]	1.607(5) [46] 1.6042(5) [40]	1.812(6) [46] 1.8137(16) [40]		2
${\displaystyle \lambda =\mu +1}$		2.1055(10) [79] 2.1056(3) [5] 2.1045(10) [80] 2.105 [81]

• For ${\displaystyle d=2}$, ${\displaystyle d_{\mathrm {b} }=2-(z^{2}-1)/12}$ where ${\displaystyle z}$ satisfies ${\displaystyle {\sqrt {3}}z/4+\sin(2\pi z/3)=0}$ near ${\displaystyle z=2.3}$. ^[6]

Exponents for protected percolation

In protected percolation, bonds are removed one at a time only from the percolating cluster. Isolated clusters are no longer modified. Scaling relations: ${\displaystyle \beta '=\beta /(1+\beta )}$, ${\displaystyle \gamma '=\gamma /(1+\beta )}$, ${\displaystyle \nu '=\nu /(1+\beta )}$, ${\displaystyle \tau '=\tau }$ where the primed quantities indicated protected percolation ^[25]

d	1	2	3	4	5	6 – ε	6 +
β'		5/41 [25]	0.288 71(15) [25]
γ'		86/41 [25]	1.3066(19) [25]
τ'		187/91 [25]	2.1659(21) [25]

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. ^{[82] [83]}

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 ${\displaystyle \zeta =1/z=\nu _{\perp }/\nu _{\parallel }}$. It satisfies the hyperscaling relation ${\displaystyle d/z=\eta +2\delta }$.

Another convention has been used for the exponent ${\displaystyle z}$, which here we call ${\displaystyle z'}$, is defined through the relation ${\displaystyle \langle R^{2}\rangle \sim t^{z'}}$, so that ${\displaystyle z'=\nu _{\parallel }/\nu _{\perp }=2/z}$. ^[84] It satisfies the hyperscaling relation ${\displaystyle dz'=2\eta +4\delta }$.

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

${\displaystyle \eta }$ (sometimes called ${\displaystyle \mu }$) is the exponent corresponding to the behavior of the average number of visited sites at time ${\displaystyle t}$ (averaged over all samples including ones that have stopped spreading): ${\displaystyle N(t)\sim t^{-\eta }}$.

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

d+1	1+1	2+1	3+1	4 – ε [85]	Mean Field
β	0.276486(8) [86] 0.276 7(3) [87]	0.5834(30) [88] 0.580(4) [87]	0.813(9) [89] 0.818(4) [87] 0.82205 [85]	${\displaystyle 1-{\frac {\varepsilon }{6}}}$	1
δ,α	0.159464(6) [86] 0.15944(2) [87]	0.4505(1) [88] 0.451(3) [84] 0.4509(5) [90] 0.4510(4) [87] 0.460(6) [91]	0.732(4) [92] 0.7398(10) [87] 0.73717 [93]	${\displaystyle 1-{\frac {\varepsilon }{4}}}$	1
η,θ	0.313686(8) [86] 0.31370(5) [87]	0.2303(4) [90] 0.2307(2) [87] 0.2295(10) [88] 0.229(3) [84] 0.214(8) [91]	0.1057(3) [87] 0.114(4) [89] 0.12084 [93]
${\displaystyle \nu _{\parallel }}$	1.733847(6) [86] 1.733825(25) [94] 1.7355(15) [87] 1.73(2) [95]	1.16(5) [95] 1.287(2) [87] 1.295(6) [84]	1.106(3) [87] 1.11(1) [89] 1.10571 [93]
${\displaystyle \nu _{\perp }}$	1.096854(4) [86] 1.096844(14) [94] 1.0979(10) [87]	0.7333(75) [92] 0.729(1) [87]	0.584(5) [92] 0.582(2) [87] 0.58360 [93]	${\displaystyle {\frac {1}{2}}+{\frac {\varepsilon }{16}}}$	${\displaystyle {\frac {1}{2}}}$
${\displaystyle z=2/z'}$	1.580745(10) [86] 1.5807(2) [87]	1.7660(16) [92] 1.765(3) [84] 1.766(2) [88] 1.7665(2) [87] 1.7666(10) [90]	1.88746 [93] 1.8990(4) [87] 1.901(5) [92]	${\displaystyle 2-{\frac {\varepsilon }{12}}}$	2
γ	2.277730(5) = 41/18?, [86] 2.278(2) [96]	1.595(18) [88]	1.237(23) [89]		1
τ	2.112(5), [97] 2.1077(13), [98] 2.10825(8) [86]

Scaling relations for directed percolation

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

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

${\displaystyle \tau =2+{\frac {2}{1+\gamma /\beta }}}$ ^[98]

${\displaystyle {\tilde {\tau }}=\nu _{\parallel }-\beta }$ ^[86]

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

${\displaystyle d_{\mathrm {DP} }=2-\beta /\nu _{\parallel }}$ ^[99]

${\displaystyle d_{b,\mathrm {DP} }=2-2\beta /\nu _{\parallel }}$ ^[99]

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

${\displaystyle dz'=2\eta +4\delta }$

${\displaystyle d/z=\eta +2\delta }$

Exponents for dynamic percolation

For dynamic percolation (epidemic growth of ordinary percolation clusters), we have

${\displaystyle P(t)\sim L^{-\beta /\nu }\sim (t^{1/d_{\mathrm {min} }})^{-\beta /\nu }=t^{-\delta }}$, implying

${\displaystyle \delta ={\frac {\beta }{\nu d_{\mathrm {min} }}}={\frac {d-d_{f}}{d_{\mathrm {min} }}}}$

For ${\displaystyle N(t)\sim t^{\eta }}$, consider ${\displaystyle N(\leq s)\sim s^{3-\tau }\sim R^{d_{f}(3-\tau )}\sim t^{d_{f}(3-\tau )/d_{\mathrm {min} }}}$, and taking the derivative with respect to ${\displaystyle t}$ yields ${\displaystyle N(t)\sim t^{d_{f}(3-\tau )/d_{\mathrm {min} }-1}}$, implying

${\displaystyle \eta ={\frac {d_{f}(3-\tau )}{d_{\mathrm {min} }}}-1={\frac {2d_{f}-d}{d_{\mathrm {min} }}}-1}$

Also, ${\displaystyle z=d_{\mathrm {min} }}$

Using exponents above, we find

d:	2	3	4	5	6 – ε	Mean Field
${\displaystyle \delta }$	0.09212	0.34681	0.59556	0.8127		1
${\displaystyle \eta ,\mu }$	0.584466	0.48725	0.30233	0.1314		0

See also

• Critical exponent
• Percolation theory
• Percolation threshold
• Percolation surface critical behavior
• Conductivity near the percolation threshold

Notes

1. For higher-order terms in the ${\displaystyle \varepsilon }$ expansions, see. ^{[18] [19] [20]}

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. Nolin, Pierre; Wei Qian; Xin Sun; Zijie Zhuang (2023). "Backbone exponent for two-dimensional percolation". arXiv: 2309.05050 [math.PR].
7. 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.
8. Binder, K.; P. C. Hohenberg (1972). "Phase Transitions and Static Spin Correlations in Ising Models with Free Surfaces". Physical Review B. 6 (9): 3461–3487. Bibcode:1972PhRvB...6.3461B. doi:10.1103/PhysRevB.6.3461.
9. De'Bell, Keith (1980). Surface effects in percolation (PhD Thesis). University of London.
10. DeBell; J. Essam (1980). "Series expansion studies of percolation at a surface". J. Phys. C: Solid State Phys. 13 (25): 4811–4821. Bibcode:1980JPhC...13.4811D. doi:10.1088/0022-3719/13/25/023.
11. Binder, K. (1981). "Critical Behaviour at Surfaces". In Domb, C.; Lebowitz, J. L. (eds.). Phase Transitions and Critical Phenomena, Volume 8. Academic Press. pp. 1–144. ISBN   978-0122203084.
12. Lubensky, T. C.; M. H. Rubin (1975). "Critical phenomena in semi-infinite systems. I. ${\displaystyle \epsilon }$ expansion for positive extrapolation length". Physical Review B. 11 (11): 4533–4546. Bibcode:1975PhRvB..11.4533L. doi:10.1103/PhysRevB.11.4533.
13. Pleimling, M (2004). "Critical phenomena at perfect and non-perfect surfaces". Journal of Physics A: Mathematical and General. 37 (19): R79–R115. arXiv: cond-mat/0402574 . doi:10.1088/0305-4470/37/19/R01. S2CID   15712212.
14. 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. Bibcode:1977JPhA...10L.203R. doi:10.1088/0305-4470/10/11/007.
15. 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. S2CID   250755965.
16. 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.
17. Priest, R. G.; T. C. Lubensky (1976). "Critical properties of two tensor models with application to the percolation problem". Physical Review B. 13 (9): 4159–4171. Bibcode:1976PhRvB..13.4159P. doi:10.1103/PhysRevB.13.4159.
18. 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 .
19. 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.
20. Borinsky, M; J. A. Gracey; M. V. Kompaniets; O. Schnetz (2021). "Five loop renormalization of phi^3 theory with applications to the Lee-Yang edge singularity and percolation theory". Physical Review D. 103: 116024. arXiv: 2103.16224 . doi:10.1103/PhysRevD.103.116024. S2CID   232417253.
21. 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.
22. Kirkpatrick, Scott (1976). "Percolation phenomena in higher dimensions: Approach to the mean-field limit". Phys. Rev. Lett. 36 (2): 69–72. Bibcode:1976PhRvL..36...69K. doi:10.1103/PhysRevLett.36.69.
23. 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.
24. 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.
25. Fayfar, Sean; Bretaña, Alex; Montfrooij, Wouter (January 15, 2021). "Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems". Journal of Physics Communications. 5 (1): 015008. arXiv: 2008.08258 . Bibcode:2021JPhCo...5a5008F. doi: 10.1088/2399-6528/abd8e9 . ISSN   2399-6528.
26. Adler, J.; Y. Meir; A. Aharony; A.B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Phys. Rev. B. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID   9993262.
27. Gaunt, D. S.; H. Ruskin (1978). "Bond percolation processes in d dimensions". J. Phys. A: Math. Gen. 11 (7): 1369–1380. Bibcode:1978JPhA...11.1369G. doi:10.1088/0305-4470/11/7/025.
28. 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.
29. 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.
30. 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.
31. 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.
32. 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.
33. 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:1975JETP...42..197L.,
34. 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.
35. 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.,
36. 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.
37. 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.
38. 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.
39. 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.
40. Zhang, Zhongjin; Pengcheng Hou; Sheng Fang; Hao Hu; Youjin Deng (2021). "Critical exponents and universal excess cluster number of percolation in four and five dimensions". Physica A: Statistical Mechanics and Its Applications. 580: 126124. arXiv: 2004.11289 . Bibcode:2021PhyA..58026124Z. doi:10.1016/j.physa.2021.126124. S2CID   216080833.
41. Tan, Xiao-Jun; Deng, You-Jin; Jacobsen, Jesper Lykke (2020). "N-cluster correlations in four- and five-dimensional percolation". Frontiers of Physics. 15 (4): 41501. arXiv: 2006.10981 . Bibcode:2020FrPhy..1541501T. doi:10.1007/s11467-020-0972-6. ISSN   2095-0462. S2CID   219956814.
42. Jan, N; Hong, D C; Stanley, H E (October 21, 1985). "The fractal dimension and other percolation exponents in four and five dimensions". Journal of Physics A: Mathematical and General. 18 (15): L935–L939. Bibcode:1985JPhA...18L.935J. doi:10.1088/0305-4470/18/15/006. ISSN   0305-4470.
43. 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.
44. 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.
45. 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.
46. 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.
47. 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.
48. 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.
49. 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.
50. 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.
51. 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.
52. 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. S2CID   121516245.
53. 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.
54. 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.
55. 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.
56. 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.
57. 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.
58. Gingold, David B.; C. J. Lobb (1990). "Percolative conduction in three dimensions". Physical Review B. 42 (13): 8220–8224. Bibcode:1990PhRvB..42.8220G. doi:10.1103/PhysRevB.42.8220. PMID   9994994.
59. 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.
60. 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.
61. 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.
62. 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.
63. 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.
64. 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.
65. 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.
66. 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.
67. Hansen, A; P. M. Lam; S. Roux (1981). "Surface order parameter in three-dimensional percolation". J. Phys. A: Math. Gen. 22 (13): 2635. doi:10.1088/0305-4470/22/13/056.
68. Deng, Youjin; H. Bl\:ote (2004). "Anisotropic limit of the bond-percolation model and conformal invariance in curved geometries". Phys. Rev. E. 69 (6): 066129. Bibcode:2004PhRvE..69f6129D. doi:10.1103/PhysRevE.69.066129. PMID   15244689.
69. Baek, Seung Ki; Petter Minnhagen; Beom Jun Kim (2010). "Surface and bulk criticality in midpoint percolation". Phys. Rev. E. 81 (4): 041108. arXiv: 1004.2622 . Bibcode:2010PhRvE..81d1108B. doi:10.1103/PhysRevE.81.041108. PMID   20481678. S2CID   18938058.
70. 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. Bibcode:1992JPhA...25L.945R. doi:10.1088/0305-4470/25/15/008.
71. 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.
72. 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.
73. 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.
74. 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.
75. 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.
76. 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 (5): L829–L833. Bibcode:1984JPhA...17L.261H. doi:10.1088/0305-4470/17/5/008. S2CID   16510317.
77. 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.
78. 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.
79. 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.
80. 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.
81. Brereton, Tim; Hirsch, Christian; Schmidt, Volker; Kroese, Dirk (December 19, 2014). "A critical exponent for shortest-path scaling in continuum percolation". Journal of Physics A: Mathematical and Theoretical. 47 (50): 505003. doi:10.1088/1751-8113/47/50/505003. ISSN   1751-8113.
82. 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.
83. 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.
84. 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.
85. 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.
86. 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.
87. 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.
88. 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.
89. 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.
90. 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.
91. 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.
92. Henkel, M.; H. Hinrichsen; S. Lŭbeck (2008). Non-equilibrium phase transitions, Vol. 1: Absorbing phase transitions. Springer, Dordrecht.
93. 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.
94. Jensen, Iwan (1996). "Low-density series expansions for directed percolation on square and triangular lattices". J. Phys. A: Math. Gen. 29 (22): 7013–7040. Bibcode:1996JPhA...29.7013J. doi:10.1088/0305-4470/29/22/007.
95. 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.
96. 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.
97. Dhar, Deepak; Mustansir Barma (1981). "Monte Carlo simulationof directed percolationon a square lattice". J. Phys. C: Solid State Phys. 14 (1): Ll-L6. Bibcode:1981JPhC...14L...1D. doi:10.1088/0022-3719/14/1/001.
98. Owczarek, A. L.; A. Rechnitzer; R. Brak; A. J. Guttmann (1997). "On the hulls of directed percolation clusters". J. Physics A: Math. Gen. 30 (19): 6679. Bibcode:1997JPhA...30.6679O. doi:10.1088/0305-4470/30/19/011.
99. Deng, Youjin; Robert M. Ziff (2022). "The elastic and directed percolation backbone". J. Phys. A: Math. Theor. 55 (24): 244002. arXiv: 1805.08201 . Bibcode:2022JPhA...55x4002D. doi:10.1088/1751-8121/ac6843. S2CID   73528075.

Further reading

• Stauffer, D.; Aharony, A. (1994), Introduction to Percolation Theory (2nd ed.), CRC Press, ISBN   978-0-7484-0253-3