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85
THE KARDARPARISIZHANG EQUATION AND UNIVERSALITY CLASS
, 2011
"... Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new univ ..."
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Cited by 97 (15 self)
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Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the KardarParisiZhang (KPZ) universality class and underlying it is, again, a continuum object – a nonlinear stochastic partial differential equation – known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact onepoint distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.
Large time asymptotics of growth models on spacelike paths I: PushASEP
, 2008
"... We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom’s model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for ..."
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Cited by 71 (32 self)
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We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom’s model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy1 and Airy2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle’s trajectory as special cases.
Fluctuation properties of the TASEP with periodic initial configuration
, 2006
"... We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and det ..."
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Cited by 67 (34 self)
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We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us [27] and here we provide a selfcontained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.
Limit process of stationary TASEP near the characteristic line
, 2009
"... The totally asymmetric simple exclusion process (TASEP) on Z with the Bernoulliρ measure as initial conditions, 0 < ρ < 1, is stationary. It is known that along the characteristic line, the current fluctuates as of order t 1/3. The limiting distribution has also been obtained explicitly. In t ..."
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Cited by 31 (10 self)
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The totally asymmetric simple exclusion process (TASEP) on Z with the Bernoulliρ measure as initial conditions, 0 < ρ < 1, is stationary. It is known that along the characteristic line, the current fluctuates as of order t 1/3. The limiting distribution has also been obtained explicitly. In this paper we determine the limiting multipoint distribution of the current fluctuations moving away from the characteristics by the order t 2/3. The main tool is the analysis of a related directed last percolation model. We also discuss the process limit in tandem queues in equilibrium.
Brownian gibbs property for airy line ensembles
"... 1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is ad ..."
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Cited by 30 (8 self)
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1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is added to each of the curves of this scaling limit, an xtranslation invariant process sometimes called the multiline Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and nonintersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers ” and “outliers”. We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property – called the Brownian Gibbs property – of being invariant under the following action. Select an index 1 ≤ k ≤ N and erase Bk on a fixed time interval (a, b) ⊆ (−N, N); then replace this erased curve with a new curve on (a, b) according to the law of a Brownian bridge between the two existing endpoints ( a, Bk(a) ) and ( b, Bk(b) ) , conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edgescaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the longstanding conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multiline Airy process, and readily yields several other interesting properties of this process. 1.
Multiple orthogonal polynomials of mixed type and nonintersecting Brownian motions
 J. Approx. Theory
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Multiple orthogonal polynomial ensembles
, 2009
"... Multiple orthogonal polynomials are traditionally studied because of their connections to number theory and approximation theory. In recent years they were found to be connected to certain models in random matrix theory. In this paper we introduce the notion of a multiple orthogonal polynomial ens ..."
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Cited by 27 (7 self)
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Multiple orthogonal polynomials are traditionally studied because of their connections to number theory and approximation theory. In recent years they were found to be connected to certain models in random matrix theory. In this paper we introduce the notion of a multiple orthogonal polynomial ensemble (MOP ensemble) and derive some of their basic properties. It is shown that Angelesco and Nikishin systems give rise to MOP ensembles and that the equilibrium problems that are associated with these systems have a natural interpretation in the context of MOP ensembles.