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Instatistical mechanics, auniversality class is a collection ofmathematical models which share a singlescale-invariant limit under the process ofrenormalization group flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular,asymptotic phenomena such ascritical exponents will be the same for all models in the class.

Some well-studied universality classes are the ones containing theIsing model or thepercolation theory at their respectivephase transition points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes will have a lower and uppercritical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical exponents stabilize and can be calculated by an analog ofmean-field theory (this dimension is 4d for Ising or for directed percolation, and 6d for undirected percolation).

Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include itsreduced temperature${\displaystyle \tau }$, itsorder parameter measuring how much of the system is in the "ordered" phase, thespecific heat, and so on.

• The exponent${\displaystyle \alpha }$ is the exponent relating the specific heat C to the reduced temperature: we have${\displaystyle C={\tau }^{-\alpha }}$. The specific heat will usually be singular at the critical point, but the minus sign in the definition of${\displaystyle \alpha }$ allows it to remain positive.
• The exponent${\displaystyle \beta }$ relates the order parameter${\displaystyle \mathrm{\Psi }}$ to the temperature. Unlike most critical exponents it is assumed positive, since the order parameter will usually be zero at the critical point. So we have${\displaystyle \mathrm{\Psi }=|\tau {|}^{\beta }}$.
• The exponent${\displaystyle \gamma }$ relates the temperature with the system's response to an external driving force, or source field. We have${\displaystyle d\mathrm{\Psi }/dJ={\tau }^{-\gamma }}$, with J the driving force.
• The exponent${\displaystyle \delta }$ relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have${\displaystyle J={\mathrm{\Psi }}^{\delta }}$ (hence${\displaystyle \mathrm{\Psi }=J^{1/\delta }}$), with the same meanings as before.
• The exponent${\displaystyle \nu }$ relates the size of correlations (i.e. patches of the ordered phase) to the temperature; away from the critical point these are characterized by acorrelation length${\displaystyle \xi }$. We have${\displaystyle \xi ={\tau }^{-\nu }}$.
• The exponent${\displaystyle \eta }$ measures the size of correlations at the critical temperature. It is defined so that thecorrelation function scales as${\displaystyle r^{-d+2-\eta }}$.
• The exponent${\displaystyle \sigma }$, used inpercolation theory, measures the size of the largest clusters (roughly, the largest ordered blocks) at 'temperatures' (connection probabilities) below the critical point. So${\displaystyle s_{max}\sim (p_c-p{)}^{-1/\sigma }}$.
• The exponent${\displaystyle \tau }$, also frompercolation theory, measures the number of sizes clusters far from${\displaystyle s_{max}}$ (or the number of clusters at criticality):${\displaystyle n_s\sim s^{-\tau }f(s/s_{max})}$, with the${\displaystyle f}$ factor removed at critical probability.

For symmetries, the group listed gives the symmetry of the order parameter. The group${\displaystyle {\mathrm{D}\mathrm{i}\mathrm{h}}_n}$ is thedihedral group, the symmetry group of then-gon,${\displaystyle S_n}$ is then-elementsymmetric group,${\displaystyle \mathrm{O}\mathrm{c}\mathrm{t}}$ is theoctahedral group, and${\displaystyle O(n)}$ is theorthogonal group inn dimensions.1 is thetrivial group.

class	dimension	Symmetry	${\displaystyle \alpha }$	${\displaystyle \beta }$	${\displaystyle \gamma }$	${\displaystyle \delta }$	${\displaystyle \nu }$	${\displaystyle \eta }$
3-statePotts	2	${\displaystyle S_3}$	⁠1/3⁠	⁠1/9⁠	⁠13/9⁠	14	⁠5/6⁠	⁠4/15⁠
Ashkin–Teller (4-state Potts)	2	${\displaystyle S_4}$	⁠2/3⁠	⁠1/12⁠	⁠7/6⁠	15	⁠2/3⁠	⁠1/4⁠
Ordinary percolation	1	1	1	0	1	${\displaystyle \mathrm{\infty }}$	1	1
	2	1	−⁠2/3⁠	⁠5/36⁠	⁠43/18⁠	⁠91/5⁠	⁠4/3⁠	⁠5/24⁠
	3	1	−0.625(3)	0.4181(8)	1.793(3)	5.29(6)	0.87619(12)	0.46(8) or 0.59(9)
	4	1	−0.756(40)	0.657(9)	1.422(16)	3.9 or 3.198(6)	0.689(10)	−0.0944(28)
	5	1	≈ −0.85	0.830(10)	1.185(5)	3.0	0.569(5)	−0.075(20) or −0.0565
	6+	1	−1	1	1	2	⁠1/2⁠	0
Directed percolation	1	1	0.159464(6)	0.276486(8)	2.277730(5)	0.159464(6)	1.096854(4)	0.313686(8)
	2	1	0.451	0.536(3)	1.60	0.451	0.733(8)	0.230
	3	1	0.73	0.813(9)	1.25	0.73	0.584(5)	0.12
	4+	1	−1	1	1	2	⁠1/2⁠	0
Conserved directed percolation (Manna, or "local linear interface")	1	1		0.28(1)		0.14(1)	1.11(2)[1]	0.34(2)[1]
	2	1		0.64(1)	1.59(3)	0.50(5)	1.29(8)	0.29(5)
	3	1		0.84(2)	1.23(4)	0.90(3)	1.12(8)	0.16(5)
	4+	1		1	1	1	1	0
Protected percolation	2	1		5/41[2]	86/41[2]
	3	1		0.28871(15)[2]	1.3066(19)[2]
Ising	2	${\displaystyle {\mathbb{Z}}_2}$	0	⁠1/8⁠	⁠7/4⁠	15	1	⁠1/4⁠
	3	${\displaystyle {\mathbb{Z}}_2}$	0.11008(1)	0.326419(3)	1.237075(10)	4.78984(1)	0.629971(4)	0.036298(2)
XY	3	${\displaystyle O(2)}$	-0.01526(30)	0.34869(7)	1.3179(2)	4.77937(25)	0.67175(10)	0.038176(44)
Heisenberg	3	${\displaystyle O(3)}$	−0.12(1)	0.366(2)	1.395(5)		0.707(3)	0.035(2)
Mean field	all	any	0	⁠1/2⁠	1	3	⁠1/2⁠	0
Molecular beam epitaxy[3]
Gaussian free field

• Universality classes from Sklogwiki
• Zinn-Justin, Jean (2002).Quantum field theory and critical phenomena, Oxford, Clarendon Press (2002),ISBN 0-19-850923-5
• Ódor, Géza (2004). "Universality classes in nonequilibrium lattice systems".Reviews of Modern Physics.76 (3):663–724.arXiv:cond-mat/0205644.Bibcode:2004RvMP...76..663O.doi:10.1103/RevModPhys.76.663.S2CID 96472311.
• Creswick, Richard J.; Kim, Seung-Yeon (1997). "Critical Exponents of the Four-State Potts Model".Journal of Physics A: Mathematical and General.30 (24):8785–8786.arXiv:cond-mat/9701018.doi:10.1088/0305-4470/30/24/036.S2CID 16687747.

• Critical phenomena
• Renormalization group
• Scale-invariant systems

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