= Percolation critical exponents
These <critical exponents> describe the leading singular powers of <percolation probability>, <percolation susceptibility>, <correlation length> and critical cluster distributions near the <percolation critical probability>. Their existence and exact values require theorems or scaling hypotheses for the particular model; they are not consequences of the definition of the threshold. For $\theta(p)=P_p(0\leftrightarrow\infty)$ and $\chi(p)=\mathbb E_p|C(0)|$, the definitions are $\theta(p)=(p-p_c)^{\beta+o(1)}$ above $p_c$ and $\chi(p)=(p_c-p)^{-\gamma+o(1)}$ below $p_c$. The <correlation length> has $\xi(p)=|p-p_c|^{-\nu+o(1)}$. At criticality, the root-cluster tail is $P(|C(0)|\geq s)=s^{-1/\delta+o(1)}$; two-point connectivity has leading power $|x|^{-(d-2+\eta)}$; and the expected number per site of size-$s$ clusters has leading power $s^{-\tau}$. The last distribution differs from the size-weighted distribution seen at a specified site. Relations such as $\gamma=(2-\eta)\nu$ and $2\beta+\gamma=d\nu$ have specific scaling regimes, and the latter <hyperscaling relation> fails in ordinary mean-field regimes above the <upper critical dimension of percolation>.
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