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69
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 101 (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 78 (35 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.
Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process
"... We consider the totally asymmetric simple exclusion process (TASEP) in discrete time with sequential update. The joint distribution of the positions of selected particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. We focus on periodic initial ..."
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Cited by 33 (19 self)
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We consider the totally asymmetric simple exclusion process (TASEP) in discrete time with sequential update. The joint distribution of the positions of selected particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. We focus on periodic initial conditions where particles occupy d�, d ≥ 2. In the proper large time scaling limit, the fluctuations of particle positions are described by the Airy1 process. Interpreted as a growth model, this confirms universality of fluctuations with flat initial conditions for a discrete set of slopes. 1
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 32 (12 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. 1 Introduction and
Transition between Airy1 and Airy2 processes and TASEP fluctuations
 Comm. Pure Appl. Math
"... We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuat ..."
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Cited by 31 (15 self)
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We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy1 and Airy2 processes, whose onepoint distributions are the GOE and GUE TracyWidom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its onepoint distribution is a new interpolation between GOE and GUE edge distributions. 1
Anisotropic growth of random surfaces in 2+1 dimensions: fluctuations and covariance
, 2009
"... We construct a family of stochastic growth models in 2+1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have ..."
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Cited by 31 (7 self)
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We construct a family of stochastic growth models in 2+1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The onepoint fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t ≫ 1. (3) There is a map of the (2+1)dimensional spacetime to the upper halfplane H such that on spacelike submanifolds the multipoint fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H. Contents 1
The universal Airy1 and Airy2 processes in the Totally Asymmetric Simple Exclusion Process
 INTEGRABLE SYSTEMS AND RANDOM MATRICES: IN HONOR OF PERCY DEIFT
, 2007
"... In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy1 and Airy2 processes. The Airy2 process is an universal limit process occurring also in other models: in a stochastic growth model on 1+1dimensions, 2d last passage percolation, equilibr ..."
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Cited by 27 (16 self)
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In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy1 and Airy2 processes. The Airy2 process is an universal limit process occurring also in other models: in a stochastic growth model on 1+1dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy1 and Airy2 processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.
A pedestrian’s view on interacting particle systems, KPZ universality, and random matrices
, 2010
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Asymptotics of Plancherel measures for the infinitedimensional unitary group
 Advances in Math
"... We study a twodimensional family of probability measures on infinite GelfandTsetlin schemes induced by a distinguished family of extreme characters of the infinitedimensional unitary group. These measures are unitary group analogs of the wellknown Plancherel measures for symmetric groups. We sh ..."
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Cited by 23 (11 self)
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We study a twodimensional family of probability measures on infinite GelfandTsetlin schemes induced by a distinguished family of extreme characters of the infinitedimensional unitary group. These measures are unitary group analogs of the wellknown Plancherel measures for symmetric groups. We show that any measure from our family defines a determinantal point process on Z+ ×Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes. 1