Critical exponent

A critical exponent describes a power-law singularity near a continuous phase transition, such as $\xi \sim |t{|}^{-\nu }$ or $\chi \sim |t{|}^{-\gamma }$.

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.

The reduced temperature is a dimensionless distance from a critical temperature. Multiplying it by a nonzero constant does not change any critical exponent.

The heat-capacity critical exponent is defined by the singular part $c_{\mathrm{s}}\sim |t{|}^{-\alpha }$ as the reduced temperature tends to zero.

The order-parameter critical exponent is defined in the ordered phase by $m\sim (-t{)}^{\beta }$ at zero conjugate field.

The magnetic-susceptibility critical exponent is defined by the zero-field magnetic susceptibility $\chi \sim |t{|}^{-\gamma }$. Separate amplitudes, and occasionally separate exponents, may occur on the two sides of the transition.

The critical-isotherm exponent is defined at the critical temperature by $m\sim |B{|}^{1/\delta }$ as the conjugate field $B$ tends to zero.

The correlation-length critical exponent is defined by $\xi \sim |t{|}^{-\nu }$. At a renormalization-group fixed point, its reciprocal is the positive eigenvalue of the temperature-like scaling direction.