nLab universality class (Rev #2)

Idea

Universality is a “term of the art” which means the following:

Under the proper conditions, different systems can exhibit the same behaviour, as measured by quantitative indices, if they meet the same qualitative criteria. Sets of systems which are equivalent in this manner are known as universality classes.

If we have some complicated phenomenon we can’t understand directly, and we figure out (or make a good stab at guessing) the universality class to which it belongs, we can make testable predictions about the complicated thing by working with a simpler member of that universality class. Membership in a universality class depends on properties like how many spatial dimensions a system lives in, symmetries and the like. People have identified universality classes, with varying degrees of rigour. Lots of them have names, sometimes even with dashes inside; the ones we understand less well and aren’t so familiar with get abstruse symbols for labels instead. A non-exhaustive tabulation of these labels might look something like this:

Stable Distributions	Equilibrium	Random Matrices	Non-Equilibrium	Extreme-Value Distributions	Dynamical Maps
Gaussian	2D Ising	Unitary	Directed percolation	Gumbel	1D Feigenbaum
Cauchy	2D Potts	Orthogonal	Dynamic percolation	Fréchet	2D Volume-preserving
Lévy	2D Tricritical Ising	Symplectic	CDP	Weibull	$\vdots$
	2D Tricritical Potts	$U(N+M)/U(N)\times U(M)$	TDP
	2D Other RSOS	$Sp(N+M)/Sp(N)\times Sp(M)$	Manna
	3D Ising	$Sp(2N)$	Edwards–Wilkinson
	3D Potts	$SO(2N)$	KPZ
	$\vdots$	$O(N+M) / O(N) \times O(M)$	$\vdots$
		$SO(2N) / U(N)$
		$Sp(2N) / U(N)$

The general concept of renormalization is really important here. The universality classes we understand best correspond to fixed points of renormalization-group transforms.

The first column, “stable distributions”, basically comes from the central limit theorem and the ways in which the conditions necessary for it to apply can fail to obtain. The middle column with all the funny symbols comes from Élie Cartan’s classification of symmetric spaces. The 2D part of the equilibrium statistical systems can be given a taxonomy based on the ADE classification of Dynkin diagrams and conformal field theories.

Relationships from one column to another do exist. For example, dynamical surface growth and random matrix theory are, unexpectedly, linked: the eigenvalue distributions of the Gaussian Unitary and Gaussian Orthogonal Ensembles show up as surface heights in Kardar-Parisi-Zhang phenomena. See Takeuchi et al. (2011), “Growing interfaces uncover universal fluctuations behind scale invariance” Sci Rep. (Nature) 1 34, arXiv:1108.2118.

References

Various sources: * T. Tao (2010), “A second draft of a non-technical article on universality” (web) * J. P. Sethna (2006), Statistical Physics: Entropy, Order Parameters, and Complexity. Oxford University Press. Available on the author’s website. See in particular chapter 12. * Nonequilibrium Phase Transitions (2008) by Henkel et al. * Vollmayr-Lee’s talks on the “field theory approach to diffusion-limited reactions”, Boulder School for Condensed Matter and Materials Physics (2009), in particular lecture 4 * Kardar’s Statistical Physics of Particles and Statistical Physics of Fields * G. Odor (2004), “Universality classes in nonequilibrium lattice systems” Rev. Mod. Phys. 76: 663. arXiv:cond-mat/0205644v7 * Henkel’s Conformal Invariance and Critical Phenomena (1999) * P. A. Pearce (1994), “Recent progress in solving A–D–E lattice models” Physica A 205: 15–30. (pdf) This one brings in Temperley-Lieb algebra?s, two-colour braid monoid algebras and Reidemeister moves. * P. Cvitanovic, ed. Universality in Chaos (1983). * and this conversation at the n-Category Café on the Tenfold Way.

Revision on August 30, 2011 at 00:14:36 by Blake Stacey See the history of this page for a list of all contributions to it.