Percolation critical exponents

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 \{\sigma,\, \tau\}\,\! motivated by the cluster size distribution, another choice is \{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 p_c\,\! for sufficiently large length scales, entailing the following asymptotic power laws: The fractal dimension d_\text{f}\,\! relates how the mass of the incipient infinite cluster depends on the radius or another length measure, M(L) \sim L^{d_\text{f}}\,\! at p=p_c\,\! and for large probe sizes, L\to\infty\,\!. Other notation: magnetic exponent y_h = D - d_f\,\! and co-dimension \Delta_\sigma = d - d_f\,\!. The Fisher exponent \tau\,\! characterizes the cluster-size distribution n_s\,\!, which is often determined in computer simulations. The latter counts the number of clusters with a given size (volume) s\,\!, normalized by the total volume (number of lattice sites). The distribution obeys a power law at the threshold, n_s \sim s^{-\tau}\,\! asymptotically as s\to\infty\,\!. The probability for two sites separated by a distance \vec r\,\! to belong to the same cluster decays as g(\vec r)\sim |\vec r|^{-2(d-d_\text{f})}\,\! or g(\vec r)\sim |\vec r|^{-d+(2-\eta)}\,\! for large distances, which introduces the anomalous dimension \eta\,\!. Also, \delta = (d + 2 - \eta)/(d - 2 + \eta) and \eta = 2 - \gamma/\nu. The exponent \Omega\,\! is connected with the leading correction to scaling, which appears, e.g., in the asymptotic expansion of the cluster-size distribution, n_s \sim s^{-\tau}(1+\text{const} \times s^{-\Omega})\,\! for s\to\infty\,\!. Also, \omega = \Omega/(\sigma \nu) = \Omega d_f. For quantities like the mean cluster size S \sim a_0 |p - p_c|^{-\gamma} (1 + a_1 (p - p_c)^{\Delta_1} +\ldots ), the corrections are controlled by the exponent \Delta_1 = \Omega\beta\delta = \omega \nu. 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 d_\text{b} (or d_\text{BB}). One also defines \xi=d-d_\text{b}. Correlations parallel and perpendicular to the surface decay as g_\parallel(\vec r)\sim |\vec r|^{2-d-\eta_\parallel}\,\! and g_\perp(\vec r)\sim |\vec r|^{2-d-\eta_\perp}\,\!. The mean size of finite clusters connected to a site in the surface is \chi_1\sim|p-p_c|^{-\gamma_1}. The mean number of surface sites connected to a site in the surface is \chi_{1,1}\sim|p-p_c|^{-\gamma_{1,1}}. == Scaling relations ==

Hyperscaling relations : \tau = \frac{d}{d_\text{f}} + 1\,\! : d_\text{f} = d - \frac{\beta}{\nu}\,\! : \eta = 2 + d - 2 d_\text{f}\,\! Relations based on {{math|{σ, τ} }} :\alpha = 2 - \frac{\tau - 1}{\sigma}\,\! :\beta = \frac{\tau - 2}{\sigma}\,\! :\gamma = \frac{3-\tau}{\sigma}\,\! :\beta+\gamma = \frac{1}{\sigma} = \nu d_f\,\! :\nu = \frac{\tau-1}{\sigma d} = \frac{2\beta+\gamma}{d}\,\! :\delta = \frac{1}{\tau-2}\,\! Relations based on {{math|{df, ν} }} :\alpha = 2 - \nu d\,\! :\beta = \nu (d - d_\text{f})\,\! :\gamma = \nu (2 d_\text{f} - d)\,\! :\sigma = \frac{1}{ \nu d_\text{f}}\,\! Conductivity scaling relations :d_w = d + \frac{t-\beta}{ \nu}\,\! :t' = d_w - d_f\,\! :\tilde d = 2 d_f/d_w Surface scaling relations : \eta_\parallel = 2 - d + 2 \beta_\mathrm{surf}/\nu\,\! :d_\mathrm{surf} = d - 1 -\beta_\mathrm{surf}/\nu\,\! : \eta_\parallel = 2\eta_\perp - \eta \, :\gamma_{1} = \nu (2-\eta_\perp)\,\! : \gamma + \nu = 2\gamma_1 - \gamma_{1,1} \,\! : x_1 = \beta_\mathrm{surf}/\nu\,\! == Exponents for standard percolation ==

• For d=2, d_\mathrm{b} = 2-(z^2-1)/12 where z satisfies \sqrt{3}z / 4 + \sin(2 \pi z/3) = 0 near z = 2.3 . == 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: \beta' = \beta/(1+\beta) , \gamma' = \gamma/(1+\beta) , \nu' = \nu/(1+\beta) , \tau' = \tau where the primed quantities indicated protected percolation == 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. \nu_\perp and \nu_\parallel are the transverse (perpendicular) and longitudinal (parallel) correlation length exponents, respectively. Also \zeta = 1/z = \nu_\perp / \nu_\parallel . It satisfies the hyperscaling relation d/z = \eta + 2 \delta . Another convention has been used for the exponent z, which here we call z', is defined through the relation \langle R^2 \rangle \sim t^{z'} , so that z'= \nu_\parallel/\nu_\perp = 2/z . d_{b,\mathrm{DP}} = 2 - 2\beta/\nu_\parallel \Delta = \beta + \gamma dz' = 2 \eta + 4 \delta d/z = \eta + 2 \delta == Exponents for dynamic percolation ==

For dynamic percolation (epidemic growth of ordinary percolation clusters), we have P(t) \sim L^{-\beta/\nu} \sim (t^{1/d_\mathrm{min}})^{-\beta/\nu} = t^{-\delta} , implying \delta = \frac{\beta}{\nu d_\mathrm{min}} = \frac{d - d_f}{d_\mathrm{min}} For N(t)\sim t^\eta, consider N(\le 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 t yields N(t)\sim t^{d_f(3-\tau)/d_\mathrm{min}-1} , implying \eta = \frac{d_f(3-\tau)}{d_\mathrm{min}}-1 = \frac{2 d_f - d}{d_\mathrm{min}}-1 Also, z = d_\mathrm{min} Using exponents above, we find == See also ==