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75
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 103 (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.
Scaling limit for the spacetime covariance of the stationary totally asymmetric simple exclusion process
 Comm. Math. Phys
"... The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. Fo ..."
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Cited by 83 (30 self)
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The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. For j = (1 − 2ρ)t + 2w(ρ(1 − ρ)) 1/3 t 2/3 we prove that the fluctuations of Nt(j) for large t are of order t 1/3 and we determine the limiting distribution function Fw(s), which is a generalization of the GUE TracyWidom distribution. The family Fw(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a RiemannHilbert problem. In our work we arrive at Fw(s) through the asymptotics of a Fredholm determinant. Fw(s) is simply related to the scaling function for the spacetime covariance of the stationary TASEP, equivalently to the asymptotic transition
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.
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 71 (36 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.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 43 (6 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
A determinantal formula for the GOE TracyWidom distribution
 J. Phys. A
"... Investigating the long time asymptotics of the totally asymmetric simple exclusion process, Sasamoto obtains rather indirectly a formula for the GOE TracyWidom distribution. We establish that his novel formula indeed agrees with more standard expressions. 1 ..."
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Cited by 36 (13 self)
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Investigating the long time asymptotics of the totally asymmetric simple exclusion process, Sasamoto obtains rather indirectly a formula for the GOE TracyWidom distribution. We establish that his novel formula indeed agrees with more standard expressions. 1
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 34 (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
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