Power laws and Critical exponents
Contents
Power laws and Critical exponents#
List of functions#
property |
function |
|---|---|
percolation probability |
\(\theta(p)=P_{p}(|C|=\infty)\) |
mean cluster size |
\(\chi(p)=E_{p}(|C|)\) |
truncated mean cluster size |
\(\chi^{\mathrm{f}}(p)=E_{p}(|C|; |C|<\infty)\) |
number of clusters per vertex |
\(\kappa(p)=E_{p}(|C|^{-1})\) |
cluster moments |
\(\chi^{\mathrm{f}}_{k}(p)=E_{p}(|C|^{k}; |C|<\infty)\) |
correlation length |
\(\xi_{p}\) |
List of critical exponents#
Near the critical point \(p_{\mathrm{c}}\)#
critical exponent |
behavior |
|---|---|
\(\beta\) |
\(\theta_{p}\approx(p-p_{\mathrm{c}})^{\beta}\) |
\(\gamma\) |
\(\chi^{\mathrm{f}}(p)\approx|p-p_{\mathrm{c}}|^{-\gamma}\) |
\(\alpha\) |
\(\kappa'''(p)\approx|p-p_{\mathrm{c}}|^{-1-\alpha}\) |
\(\Delta\) |
\(\dfrac{\chi^{\mathrm{f}}_{k+1}(p)}{\chi^{\mathrm{f}}_{k}(p)}\approx|p-p_{\mathrm{c}}|^{-\Delta}\) |
\(\nu\) |
\(\xi(p)\approx |p-p_{\mathrm{c}}|^{-\nu}\) |
On the critical point \(p_{\mathrm{c}}\)#
critical exponent |
behavior |
|---|---|
\(\delta\) |
\(P_{p_{\mathrm{c}}}(|C|=n)\approx n^{-1-1/\delta}\) |
\(\rho\) |
\(P_{p_{\mathrm{c}}}(\mathrm{rad}(C)=n)\approx n^{-1-1/\rho}\) |
\(\eta\) |
\(P_{p_{\mathrm{c}}}(0\leftrightarrow x)\approx|x|^{2-d-\eta}\) |