Universal Phenomena In Random Systems - Columbia University
Universal phenomena in random systemsIvan Corwin (Clay Mathematics Institute, Columbia University, Institute Henri Poincare)Tuesday talk 1 Page 1
Integrable probabilistic systems Admit exact and concise formulas for expectations of a variety ofobservables of interest. Asymptotics of systems, observables and formulas lead to detaileddescriptions of wide universality classes and limiting phenomena.These special systems come from algebraic structures:Representation theoryQuantum integrable systems(Macdonald processes)(stochastic vertex models)Integrableprobabilistic systemsUniversality of asymptotics requires different tools - mostly open!Tuesday talk 1 Page 2
Coin flipping The number of heads in N fair coin flips isgiven exactly by the Binomial distribution:( )Probability (heads h) 2-N Law of large numbers [Bernoulli 1713]: as N grows, heads/N - 1/2 Central limit theorem [de Moivre 1733], [Laplace 1812]: as N growsProbability (heads ) - Proved using asymptotics for N! [de Moivre 1721], [Stirling 1729]:Tuesday talk 1 Page 3
The Gaussian central limit theoremThe universality of the Gaussian distributionwas not demonstrated until [Lyapunov 1901].Polya called this the 'central limit theorem'due to its importance in probability theory.Theorem: Let be independent identically distributed (iid) randomvariables of finite mean and variance . Then for all , as growsProbability () - Extensions exist for this result, and much ofprobability deals with Gaussian processes. The 'bell curve' is ubiquitous and is the basisfor much of classical statistics.Tuesday talk 1 Page 4
Random deposition modelBlocks fall independent andin parallel above each siteaccording to exponentiallydistributed waiting times.Exponential distribution of rate λ (mean 1/λ):Probability (X s) -λse .Memoryless (Markov), so growth depends only the present state.Gaussian behavior since each column is a sum of iid random variablesTuesday talk 1 Page 5
Ballistic deposition model (sticky blocks)Same process offalling blocksSticky blocks introduce partial correlation [Vold 1959]Tuesday talk 1 Page 6
Random vs. ballistic depositionGaussian universal classKPZ universality class Linear growth (known speed)1/2 tfluctuations with Linear growth (unknown speed)1/3 Conjectural tGaussian limit (CLT) No spatial correlationfluctuations withGOE Tracy-Widom limit. Conjectural t2/3 spatial correlationTuesday talk 1 Page 7
Ballistic deposition in 'nature'Tuesday talk 1 Page 8
Disordered liquid crystal growthTuesday talk 1 Page 9
Corner growth model - an integrable exampleEachturns intoafter anexponential rate 1 waiting time.h(t,x) height above x at time t. Wedge initial data is h(0,x) x .Theorem [Rost 1981]: For wedge initial data asTuesday talk 1 Page 10grows,
Corner growth model - an integrable exampleL2/3L1/3Eachturns intoLafter anexponential rate 1 waiting time.Define the rescaled height functionTheorem [Johansson 1999]: For wedge initial data asProbability( Tuesday talk 1 Page 11) - grows.
GUE Tracy-Widom distribution (FGUE or F2) First arose in the study of randommatrices [Tracy-Widom 1993] Negative mean, lower tail likeProbabilitydensityfunction p(x)and upper tail like Defined via a Fredhold determinant:Log[p(x)]Tuesday talk 1 Page 12
1 1 dimensional Kardar-Parisi-Zhang universality class Entire growth processes has a limit - the KPZ fixed point. 3 : 2 : 1 scaling of time : space : fluctuation is called 'KPZ scaling'. Believed to arise in 1 1 dimensional growth processes which enjoy Local dynamics Smoothing Slope dependent (or lateral) growth rate Space-time random driving forces There are a number of other types of systems which can (at leastin special cases or approximations) be maps into growth processes.Hence these become included into the universality class too.Tuesday talk 1 Page 13
Filling in the KPZ universality classRandom interface growthTraffic flowStochastic PDEsKPZfixed pointBig data andrandom matricesRandom tilingsOptimal paths / random walksin random environmentKPZ fixed point should be the universal limit under 3:2:1 scaling.This is mainly conjectural and only proved for integrable models.Tuesday talk 1 Page 14
Random interface growth Partially asymmetric corner growth model: Eachturns intoafter anexponential rate p waiting time. Eachturns intop qafter anexponential rate q waiting time.Theorem [Tracy-Widom '09]: Same law of large numbersand fluctuation limit theorems hold with t - t/(p-q).When p q the law of large numbers and fluctuations change nature.This corresponds with the Edwards-Wilkinson universality classwhich has 4:2:1 scaling and Gaussian limiting behavior.Tuesday talk 1 Page 15
Stochastic partial differential equations: KPZ equationspace-time white noise Continuum growth model studied by [Kardar-Parisi-Zhang '86]using work of [Forster-Nelson-Stephen '77] to predict 3:2:1 scaling. [Bertini-Cancrini '95], [Bertini-Giacomin '97] make sense of this. KPZ equation is in the KPZ universality class proved recently: 3:2:1 scaling [Balazs-Quastel-Seppalainen '09] FGUE limit [Amir-C-Quastel '10]EW (4:2:1)Short timefixed pointKPZ equationlong timefixed pointCorner growthp-q 0p-q - 0 criticallyTuesday talk 1 Page 16KPZ (3:2:1)p-q 0 fixed
Rescaling the KPZ equationThe rescaled solutionsatisfiesKPZ scaling: b 1/2, z 3/2 [Forster-Nelson-Stephen '77], [KPZ '86]All growth processes with key features (locality, smoothing, lateralgrowth, noise) should renormalize to KPZ fixed point [C-Quastel '11](e.g. GUE Tracy-Widom law). Unclear exactly what this limit is!Weak nonlinearity scaling: b 1/2, z 2, scale nonlinearity byWeak noise scaling: b 0, z 2, scale noise by.Weak limits are proxies for rescaling discrete models to KPZ equation.Tuesday talk 1 Page 17
Another big pictureWeak nonlinearityASEPRate qq-TASEPRateTASEPscalingRate pKPZ equationWeak noisescalingKPZ scalingKPZ fixed pointTuesday talk 1 Page 181 1 dimensionalsemi-discrete anddiscrete SHEDirected polymers
Traffic flow Asymmetric simple exclusion process (ASEP):Introduced in biology literature to model RNAtranscription [MacDonald-Gibbs-Pipkin '68]. q-TASEP [Borodin-C '11]: Simple traffic modelrate 1-qgapgap 4 q-PushASEP [C-Petrov '12]: Includes breakingprobability qgaprate 1rate 1-qgapgap 2gap 4KPZ class behavior: For step initial data, the number of particles tocross origin behaves likewhereTuesday talk 1 Page 19is FGUE distributed.
Optimal paths in random environmentp 1, q 0Last passage percolation [Rost '81]: time for box (i,j) to grow, once it can (exponential rate 1). L(x,y): time when box x,y is grown.Recursion: L(x,y) max(L(x-1,y),L(x,y-1)) wxyIterating: L(x,y) maxwijKPZ class behavior: L(xt,yt) behaves likeFGUE distributed and the constants depend on x,y.Tuesday talk 1 Page 20whereis
Optimal paths in random environment[Barraquand-C '15]: Assign edge weights to eachso with probability1/2, horizontal weight is 0 and vertical is exp(1); otherwise reversed.Minimal passage time P(x,y) minwe .KPZ class behavior: For x y, P(xt,yt) behaves likewhereis FGUE distributed and the constants depend on x,y.Tuesday talk 1 Page 21
Random walk in random environmentspaceFor each (space,time)-vertextimechoose uys uniform on [0,1].Take independent random walks X(1),X(2) , where at time s and position y,move left with probability uys, right with1-uys. Let M(t,N) max ( X(1), , X(N) ).rtKPZ class behavior: For 0 r 1, M(t,e ) behaves likewhereis FGUE and the constants depend on r [Barraquand-C '15].If all uys 1/2 (i.e. simple symmetric random walk), large deviations andextreme value theory implies order one Gumbel fluctuations.Tuesday talk 1 Page 22
Big data and random matricesGaussian Unitary Ensemble (GUE) on N x N complex matrices:whereIntroduced by [Wigner '55] to model the energy levels/gaps of atomstoo complicated to solve analytically.Letdenote the(random) real eigenvalues ofKPZ class behavior:.behaves likewhereis FGUE.Relationship to growth processes is much less apparent here.Tuesday talk 1 Page 23
Big data and random matricesComplex Wishart Ensemble (or sample covariance) on N x M matrices:whereIntroduced by [Wishart '28] within statistics. Provides a base-line fornoisy data against which to compare Principal Component AnalysisLetofdenote the (random) real positive singular values(i.e., the square-roots of eigenvalues ofSurprise [Johansson '00]: The distribution ofE.G. N M 1, Probability()Tuesday talk 1 Page 24).equals that of L(N,M).
Vicious walkers and random tilingsConsider N random walks with fixed starting and ending points,conditioned not to touch. This gives rise to a uniform measure onfillings of a box, or tilings of a hexagon by three types of rhombi.Tuesday talk 1 Page 25
Vicious walkers and random tilingsArctic circle theorem [Cohn-Larsen-Propp '98]KPZ class behavior: The topwalker (or edge of the arcticcircle) has fluctuations of order1/3Nand limiting FGUE distribution.[Baik-Kriecherbauer-McLaughlinMiller '07], [Petrov '12]Pretty pictures (and math) whentiling various types of domainsTuesday talk 1 Page 26
Open problems Higher dimension (e.g. random surface growth) KPZ universality (scale, distribution, entire space-time limit) Growth processes (e.g. ballistic deposition, Eden model) Interacting particle systems (e.g. non-nearest neighbor exclusion) Last/first passage percolation, RWRE with general weights Full description of KPZ fixed point Complete space-time multpoint distribution Unique characterization of fixed point Weak universality of the KPZ equation Under critical weak tuning of the strength of model parameters Discover new integrable examples and tools in their analysesTuesday talk 1 Page 27
Summary Integrable probabilistic systems reveal details of large universalityclasses. They are intimately connected to certain algebraic structure Coin flipping and Gaussian universality class is simplest example Random interface growth leads to new phenomena such as spatialcorrelation, smaller fluctuations and new distributions KPZ class arises in various growing interfaces, and the analysis ofthe corner growth model reveals its properties KPZ class encompasses many other types of systems, includingstochastic PDEs, traffic flow, optimal paths in randomenvironments, random walks in random environments, big dataand random matrices, vicious walkers and tilings Tuesday talk 1 Page 28
Random interface growth Stochastic PDEs Big data and random matrices Traffic flow Random tilings in random environment Optimal paths / random walks KPZ fixed point should be the universal limit under 3:2:1 scaling. This is mainly conjectural and only proved for integrable models. KPZ fixed point Tuesday talk 1 Page 14
Start by finding out how Python generates random numbers. Type ?random to find out about scipy's random number generators. Try typing 'random.random()' a few times. Try calling it with an integer argument. Use 'hist' (really pylab.hist) to make a histogram of 1000 numbers generated by random.random. Is th
Start by finding out how Python generates random numbers. Type ?random to find out about scipy's random number generators. Try typing 'random.random()' a few times. Try calling it with an integer argument. Use 'hist' (really pylab.hist) to make a histogram of 1000 numbers generated by random.random. Is the distribution Gaussian, uniform, or .
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ONE-DIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK Definition 1. A random walk on the integers Z with step distribution F and initial state x 2Z is a sequenceSn of random variables whose increments are independent, identically distributed random variables i with common distribution F, that is, (1) Sn
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