Percolation critical exponents

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 ${\displaystyle N^{\tilde {d}}}$.

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, ${\displaystyle t'=\zeta }$.

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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