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 $P(|C(0)|\ge s)=s^{-1/\delta +o(1)}$; 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.

The mean-field predictions model large clusters by branching processes. Rigorous results apply in specified high-dimensional or sufficiently spread-out regimes, using tools such as lace expansion and the percolation triangle condition. The full list involves different observables; the triangle condition alone should not be presented as proving every spatial exponent. With these exponents $2\beta +\gamma =3$, whereas $d\nu =d/2$, explaining the failure of naive hyperscaling relation above six.

The critical cluster mass-radius scaling exponent $D_f$ describes clusters of mass roughly $r^{D_f}$ at radius $r$. The displayed relation is the below-upper-critical-dimension scaling prediction, not an identity from the definition. In the planar scaling description $D_f=91/48$.

The exponent $\delta$ describes the tail of the critical cluster seen from a specified vertex of a graph. If the expected number of size-$s$ clusters per site scales as $s^{-\tau }$, weighting by cluster size gives $1/\delta =\tau -2$ when these scaling laws hold.

The leading power $\beta$ of the percolation probability approached from above the percolation critical probability. It quantifies how the density of the infinite percolation cluster first becomes positive.