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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <blockquote> <p>This entry is about the phenomenon of <a class="existingWikiWord" href="/nlab/show/universality">universality</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. See also <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> for a (different) concept of a similar name in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>.</p> </blockquote> <hr /> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#timeline'>Timeline</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <strong>universality</strong> is a term which means the following:</p> <div class="standout"> <p>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 <strong>universality classes</strong>.</p> </div> <p>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 <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a> a system lives in, <a class="existingWikiWord" href="/nlab/show/symmetry">symmetries</a> 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:</p> <table><thead><tr><th>Stable Distributions</th><th>Equilibrium</th><th>Random Matrices</th><th>Non-Equilibrium</th><th>Extreme-Value Distributions</th><th>Dynamical Maps</th></tr></thead><tbody><tr><td style="text-align: left;">Gaussian</td><td style="text-align: left;">2D Ising</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unitary+group">Unitary</a></td><td style="text-align: left;">Directed percolation</td><td style="text-align: left;">Gumbel</td><td style="text-align: left;">1D Feigenbaum</td></tr> <tr><td style="text-align: left;">Cauchy</td><td style="text-align: left;">2D Potts</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orthogonal+group">Orthogonal</a></td><td style="text-align: left;">Dynamic percolation</td><td style="text-align: left;">Fréchet</td><td style="text-align: left;">2D Volume-preserving</td></tr> <tr><td style="text-align: left;">Lévy</td><td style="text-align: left;">2D Tricritical Ising</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+group">Symplectic</a></td><td style="text-align: left;">CDP</td><td style="text-align: left;">Weibull</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;">2D Tricritical Potts</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(N+M)/U(N)\times U(M)</annotation></semantics></math></td><td style="text-align: left;">TDP</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;">2D Other RSOS</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Sp</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Sp</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sp(N+M)/Sp(N)\times Sp(M)</annotation></semantics></math></td><td style="text-align: left;">Manna</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;">3D Ising</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sp(2N) </annotation></semantics></math></td><td style="text-align: left;">Edwards–Wilkinson</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;">3D Potts</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SO(2N) </annotation></semantics></math></td><td style="text-align: left;">KPZ</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> O(N+M) / O(N) \times O(M) </annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;"> </td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SO(2N) / U(N) </annotation></semantics></math></td><td style="text-align: left;"> </td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"> </td><td style="text-align: left;"> </td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sp(2N) / U(N) </annotation></semantics></math></td><td style="text-align: left;"> </td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>The general concept of <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> is really important here. The universality classes we understand best correspond to fixed points of renormalization-group transforms.</p> <p>The first column, “stable distributions”, basically comes from the <a class="existingWikiWord" href="/nlab/show/central+limit+theorem">central limit theorem</a> 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 <a class="existingWikiWord" href="/nlab/show/%C3%89lie+Cartan">Élie Cartan</a>‘s classification of <a class="existingWikiWord" href="/nlab/show/symmetric+space">symmetric spaces</a>. The 2D part of the <a class="existingWikiWord" href="/nlab/show/equilibrium">equilibrium</a> statistical systems can be given a taxonomy based on the <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a> of <a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagrams</a> and <a class="existingWikiWord" href="/nlab/show/conformal+field+theories">conformal field theories</a>. As <a href="#Cardy2010">Cardy (2010)</a> writes,</p> <div class="standout"> <p>[T]he study of the <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> of the <a class="existingWikiWord" href="/nlab/show/Virasoro+algebra">Virasoro algebra</a> gives a way of classifying all possible <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>s and thereby universality classes in 2d. The breakthrough in this direction came following the seminal 1984 paper of Belavin, Polyakov and Zamolodchikov (BPZ) in which they showed that, for certain special rational values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo><</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c \lt 1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> closes with only a finite number of <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s of the Virasoro algebra, and, for, these cases, all the <span class="newWikiWord">critical exponent<a href="/nlab/new/critical+exponent">?</a></span>s and multi-point <a class="existingWikiWord" href="/nlab/show/correlation+function">correlation function</a>s are calculable. Shortly thereafter Friedan, Qiu and Shenker showed that unitary CFTs (corresponding to local, positive definite Boltzmann weights) are a subset of this list, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>=</mo><mn>1</mn><mo>−</mo><mn>6</mn><mo stretchy="false">/</mo><mi>m</mi><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c = 1 - 6/m(m + 1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> an integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\geq 3</annotation></semantics></math>. This gives rise to what might be termed the ‘conformal periodic table’. The first few examples may be identified with well-known universality classes. The ‘hydrogen atom’ of CFT is the scaling limit of the critical Ising model, ‘helium’ is the tricritical Ising model, and so on. Note, however, that at the next value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>=</mo><mn>4</mn><mo stretchy="false">/</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">c = 4/5</annotation></semantics></math> two possible ‘isotopes’ arise. In the second, corresponding to the critical 3-state Potts model, not all the scaling dimensions allowed by BPZ in fact occur, but some of those that do actually appear twice. In fact the constraint of unitarity is not sufficient to determine exactly which representations actually occur in a given CFT. The answer to this is provided by demanding consistency of the theory on a <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, by interchanging the interpretations of space and imaginary time []. For the torus, this is a <a class="existingWikiWord" href="/nlab/show/modular+form">modular</a> transformation, and the requirement of modular invariance has become another powerful tool in classifying CFTs, completely solved in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo><</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c \lt 1</annotation></semantics></math> by Cappelli, Itzykson and Zuber.</p> </div> <p>Some relationships between columns can be identified. For example, dynamical surface growth and <a class="existingWikiWord" href="/nlab/show/random+matrix+theory">random matrix theory</a> are, unexpectedly, linked: the <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a> distributions of the <span class="newWikiWord">Gaussian Unitary<a href="/nlab/new/Gaussian+Unitary+Ensemble">?</a></span> and <span class="newWikiWord">Gaussian Orthogonal Ensembles<a href="/nlab/new/Gaussian+Orthogonal+Ensemble">?</a></span> show up as surface heights in Kardar-Parisi-Zhang phenomena. See (<a href="#Takeuchi">Takeuchi <em>et al.</em></a>).</p> <h2 id="timeline">Timeline</h2> <p>The modern terminology of this subject dates to the late 1960s and early ’70s. By 1968, the quantitative concurrence of <span class="newWikiWord">critical exponent<a href="/nlab/new/critical+exponent">?</a></span>s experimentally observed in magnetic materials and fluid phase transitions was sufficiently well established that it was reasonable to ask, “To what extent are the observed values of the critical-point exponents universal?” (<a href="#Fisher1968">Fisher 1968</a>; for a more recent presentation of relevant empirical figures, see <a href="#Sethna2006">Sethna 2006</a>, p. 267). At the 1970 Enrico Fermi summer school in Varenna, Italy, Leo P. Kadanoff summarized recent results with a “hypothesis of universality” which the conference proceedings record as follows:</p> <div class="standout"> <p>All phase transition problems can be divided into a small number of different classes depending upon the dimensionality of the system and the symmetries of the order state. Within each class, all phase transitions have identical behaviour in the critical region, only the names of the variables are changed. (<a href="#Kadanoff1971">Kadanoff 1971</a>)</p> </div> <p>The following is a capsule history tabulating some important events in the historical development of the modern understanding of universality.</p> <table><thead><tr><th>Year</th><th></th><th>Contributors</th><th></th><th>Event</th></tr></thead><tbody><tr><td style="text-align: left;">1733</td><td style="text-align: left;"></td><td style="text-align: left;">de Moivre</td><td style="text-align: left;"></td><td style="text-align: left;">Suggestion of what we now know as the <a class="existingWikiWord" href="/nlab/show/central+limit+theorem">central limit theorem</a></td></tr> <tr><td style="text-align: left;">1822</td><td style="text-align: left;"></td><td style="text-align: left;">Cagniard de la Tour</td><td style="text-align: left;"></td><td style="text-align: left;">Observation of critical point at end of liquid-gas coexistence curve</td></tr> <tr><td style="text-align: left;">1873–75</td><td style="text-align: left;"></td><td style="text-align: left;">van der Waals, Maxwell</td><td style="text-align: left;"></td><td style="text-align: left;">Equation of state for non-ideal gases, and its application to phase transitions</td></tr> <tr><td style="text-align: left;">1893</td><td style="text-align: left;"></td><td style="text-align: left;">van der Waals</td><td style="text-align: left;"></td><td style="text-align: left;">Use of critical exponents to characterize a phase transition (<a href="#LeveltSengers1976">Levelt Sengers 1976</a>)</td></tr> <tr><td style="text-align: left;">1895</td><td style="text-align: left;"></td><td style="text-align: left;">P. Curie</td><td style="text-align: left;"></td><td style="text-align: left;">Experimental observation of demagnetization by heat; pointing out similarity between liquid-gas and ferromagnetic transitions</td></tr> <tr><td style="text-align: left;">1896–1900</td><td style="text-align: left;"></td><td style="text-align: left;">Verschaffelt</td><td style="text-align: left;"></td><td style="text-align: left;">Discovery that the critical exponents predicted from van der Waals theory do not match experiment (<a href="#LeveltSengers1976">Levelt Sengers 1976</a>)</td></tr> <tr><td style="text-align: left;">1901</td><td style="text-align: left;"></td><td style="text-align: left;">Lyapunov</td><td style="text-align: left;"></td><td style="text-align: left;">Demonstration of why the central limit theorem works</td></tr> <tr><td style="text-align: left;">1908</td><td style="text-align: left;"></td><td style="text-align: left;">Smoluchowski</td><td style="text-align: left;"></td><td style="text-align: left;">Discovery that critical opalescence is due to density fluctuations across many scales</td></tr> <tr><td style="text-align: left;">1937</td><td style="text-align: left;"></td><td style="text-align: left;">Landau</td><td style="text-align: left;"></td><td style="text-align: left;">General mean-field theory of phase transitions</td></tr> <tr><td style="text-align: left;">1944</td><td style="text-align: left;"></td><td style="text-align: left;">Onsager</td><td style="text-align: left;"></td><td style="text-align: left;">Exact solution of 2D <a class="existingWikiWord" href="/nlab/show/Ising+model">Ising model</a>, illuminating the breakdown of analyticity and providing a valuable comparison point for later approximation techniques</td></tr> <tr><td style="text-align: left;">1947–49</td><td style="text-align: left;"></td><td style="text-align: left;">Feynman, Schwinger, Tomonaga</td><td style="text-align: left;"></td><td style="text-align: left;">Workable <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> techniques for “sweeping infinities under the rug” in <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a></td></tr> <tr><td style="text-align: left;">1954–56</td><td style="text-align: left;"></td><td style="text-align: left;">Stueckelberg, Gell-Mann, Bogoliubov <em>et al.</em></td><td style="text-align: left;"></td><td style="text-align: left;">Discovery that the “renormalization group” (RG) captured an important formal property of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></td></tr> <tr><td style="text-align: left;">1955</td><td style="text-align: left;"></td><td style="text-align: left;">Wigner</td><td style="text-align: left;"></td><td style="text-align: left;">Eigenvalue statistics of random matrices</td></tr> <tr><td style="text-align: left;">1970–72</td><td style="text-align: left;"></td><td style="text-align: left;">Kadanoff, Wilson <em>et al.</em></td><td style="text-align: left;"></td><td style="text-align: left;">Development of RG concept, discovery that it explains universal properties of phase transitions like critical exponents</td></tr> <tr><td style="text-align: left;">1976</td><td style="text-align: left;"></td><td style="text-align: left;">Doi</td><td style="text-align: left;"></td><td style="text-align: left;">Second-quantization formalism for classical stochastic many-particle dynamics</td></tr> <tr><td style="text-align: left;">1976–79</td><td style="text-align: left;"></td><td style="text-align: left;">Feigenbaum</td><td style="text-align: left;"></td><td style="text-align: left;">Observation and use of RG to explain universal scaling exponents in 1D dynamical maps</td></tr> <tr><td style="text-align: left;">1977</td><td style="text-align: left;"></td><td style="text-align: left;">Hohenberg, Halperin</td><td style="text-align: left;"></td><td style="text-align: left;">Classification of critical phenomena in dynamic continuum theories</td></tr> <tr><td style="text-align: left;">1978–80</td><td style="text-align: left;"></td><td style="text-align: left;">Grassberger, Sundermeyer, de la Torre, Cardy, Sugar</td><td style="text-align: left;"></td><td style="text-align: left;">Connection of <span class="newWikiWord">Reggeon field theory<a href="/nlab/new/Reggeon+field+theory">?</a></span> with directed percolation</td></tr> <tr><td style="text-align: left;">1980</td><td style="text-align: left;"></td><td style="text-align: left;">Libchaber <em>et al.</em></td><td style="text-align: left;"></td><td style="text-align: left;">Observation of Feigenbaum’s constants in experimental systems, starting with convection of mercury</td></tr> <tr><td style="text-align: left;">1981–82</td><td style="text-align: left;"></td><td style="text-align: left;">Janssen, Grassberger</td><td style="text-align: left;"></td><td style="text-align: left;">Formulation of DP conjecture, setting out the likely criteria for phase transitions to belong to the directed-percolation class</td></tr> <tr><td style="text-align: left;">1984</td><td style="text-align: left;"></td><td style="text-align: left;">Belavin, Polyakov, Zamolodchikov</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> in 2D</td></tr> <tr><td style="text-align: left;">1986</td><td style="text-align: left;"></td><td style="text-align: left;">Zamolodchikov</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/c-theorem">c-theorem</a> for 2D theories, identifying RG fixed points with CFTs of specific <a class="existingWikiWord" href="/nlab/show/Virasoro+algebra">central charges</a></td></tr> <tr><td style="text-align: left;">1986</td><td style="text-align: left;"></td><td style="text-align: left;">Kardar, Parisi, Zhang</td><td style="text-align: left;"></td><td style="text-align: left;">KPZ equation for dynamical surface growth by deposition, showing that RG is applicable to its far-from-equilibrium dynamics</td></tr> <tr><td style="text-align: left;">1996–97</td><td style="text-align: left;"></td><td style="text-align: left;">Zirnbauer, Altland</td><td style="text-align: left;"></td><td style="text-align: left;">Symmetry classes (not <em>exactly</em> the same thing as universality classes, but clearly part of the same picture) for random matrices, following Cartan’s classification of symmetric spaces</td></tr> <tr><td style="text-align: left;">1998–99</td><td style="text-align: left;"></td><td style="text-align: left;">Freedman, <em>et al.</em></td><td style="text-align: left;"></td><td style="text-align: left;">First indications that the <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a> and related implementations of gauge/gravity duality could lead to a “holographic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>-theorem” generalising Zamolodchikov’s work to higher dimensions</td></tr> </tbody></table> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/statistical+physics">statistical physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamic+limit">thermodynamic limit</a></p> </li> </ul> <h2 id="references">References</h2> <p>General sources:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Terence+Tao">T. Tao</a> (2010), “A second draft of a non-technical article on universality” (<a href="http://terrytao.wordpress.com/2010/09/14/a-second-draft-of-a-non-technical-article-on-universality/">web</a>)</li> </ul> <p>Statistical distributions:</p> <ul> <li>Wikipedia, <a href="http://en.wikipedia.org/wiki/Stable_distribution">stable distribution</a></li> <li>Wikipedia, <a href="http://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem">Fisher–Tippett–Gnedenko theorem</a></li> </ul> <p>Equilibrium critical phenomena:</p> <ul> <li>B. Berche <em>et al.</em> (2009), “Critical phenomena: 150 years since Cagniard de la Tour” <em>J. Phys. Studies</em> <strong>13</strong>: 3201, <a href="http://arxiv.org/abs/0905.1886">arXiv:0905.1886</a>.</li> <li>J. P. Sethna (2006), <em>Statistical Physics: Entropy, Order Parameters, and Complexity</em>. Oxford University Press. <a href="http://pages.physics.cornell.edu/sethna/StatMech/">Available on the author’s website</a>. See in particular chapter 12.</li> <li>Kardar’s <em>Statistical Physics of Particles</em> and <em>Statistical Physics of Fields</em></li> <li>Henkel’s <em>Conformal Invariance and Critical Phenomena</em> (1999)</li> <li>P. A. Pearce (1994), “Recent progress in solving <em>A–D–E</em> lattice models” <em>Physica A</em> <strong>205</strong>: 15–30. (<a href="http://mac0916.ms.unimelb.edu.au/~pap/Publications_pdf/1994PearceRecentProgressADE.pdf">pdf</a>) This one brings in <span class="newWikiWord">Temperley-Lieb algebra<a href="/nlab/new/Temperley-Lieb+algebra">?</a></span>s, two-colour braid monoid algebras and <a class="existingWikiWord" href="/nlab/show/Reidemeister+moves">Reidemeister moves</a>.</li> <li>J. Cardy (2010), <em>The Ubiquitous ‘c’: From the Stefan–Boltzmann Law to Quantum Information</em>, J.Stat.Mech. <strong>2010</strong>: P10004, (<a href="http://arxiv.org/abs/1008.2331">arXiv:1008.2331</a>)</li> <li>A. Cappelli and J.-B. Zuber (2010), <em>A-D-E Classification of Conformal Field Theories</em> Scholarpedia, 5(4): 10314 (<a href="http://www.scholarpedia.org/article/A-D-E_Classification_of_Conformal_Field_Theories">web</a>). Also available as <a href="http://arxiv.org/abs/0911.3242">arXiv:0911.3242</a>.</li> <li>Michael E. Fisher (1968), <em>Renormalization of Critical Exponents by Hidden Variables</em> Physical Review <strong>176</strong>: 257–72. (<a href="http://link.aps.org/doi/10.1103/PhysRev.176.257">web</a>)</li> <li>L. P. Kadanoff (1971), <em>Critical Behavior. Universality and Scaling</em> in Proceedings of the International School of Physics Enrico Fermi, Course LI (27 July – 8 August 1970). Edited by M. S. Green.</li> <li>J. M. H. Levelt Sengers (1976), <em>Critical Exponents at the Turn of the Century</em> Physica A <strong>82</strong>: 319–51. (<a href="http://www.sciencedirect.com/science/article/pii/0378437176900121">web</a>)</li> </ul> <p>Random matrix theory:</p> <ul> <li>Yan Fyodorov (2011) Random matrix theory. Scholarpedia, 6(3):9886 (<a href="http://www.scholarpedia.org/article/Random_matrix_theory">web</a>)</li> <li>this conversation at the n-Category Café on <a href="http://golem.ph.utexas.edu/category/2010/12/the_threefold_way_part_2.html#c035973">the Tenfold Way</a>.</li> </ul> <p>Nonequilibrium critical phenomena:</p> <ul> <li>P. C. Hohenberg and B. I. Halperin (1977). “Theory of dynamic critical phenomena” <em>Rev. Mod. Phys.</em> <strong>49</strong>: 435. (<a href="http://www.physics.mcgill.ca/~omid/Hohenberg_Halperin.pdf">web</a>)</li> <li><em>Nonequilibrium Phase Transitions</em> (2008) by Henkel <em>et al.</em></li> <li>Vollmayr-Lee’s talks on the “field theory approach to diffusion-limited reactions”, <a href="http://boulderschool.yale.edu/2009/boulder-school-2009-lecture-notes">Boulder School for Condensed Matter and Materials Physics</a> (2009), in particular lecture 4</li> <li>G. Odor (2004), “Universality classes in nonequilibrium lattice systems” <em>Rev. Mod. Phys.</em> <strong>76</strong>: 663. <a href="http://arxiv.org/abs/cond-mat/0205644">arXiv:cond-mat/0205644v7</a></li> <li>I. Corwin (2011), “The Kardar-Parisi-Zhang equation and universality class” <a href="http://arxiv.org/abs/1106.1596">arXiv:1106.1596</a>.</li> <li>Takeuchi <em>et al.</em> <em>Growing interfaces uncover universal fluctuations behind scale invariance</em> Sci Rep. (Nature) <strong>1</strong> 34, (<a href="http://arxiv.org/abs/1108.2118">arXiv:1108.2118</a>)</li> </ul> <p>Dynamical maps and chaos theory:</p> <ul> <li>P. Cvitanovic, ed. <em>Universality in Chaos</em> (1983).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 19, 2019 at 16:22:49. 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