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20
Large time asymptotics of growth models on spacelike paths II: PNG and parallel TASEP, in preparation
, 2007
"... 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 19 (11 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. 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 11 (7 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
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 10 (5 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
The universal Airy1 and Airy2 processes in the Totally Asymmetric Simple Exclusion
 Process, Integrable Systems and Random Matrices: In Honor of Percy Deift
"... 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 8 (7 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. 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 8 (5 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
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 8 (1 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.
2008, The Airy 1 process is not the limit of the largest eigenvalue in GOE matrix diffusion
"... Abstract Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion. ..."
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Cited by 5 (3 self)
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Abstract Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion.
Determinantal Correlations for Classical Projection Processes
, 801
"... Recent applications in queuing theory and statistical mechanics have isolated the process formed by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in general from the eigenvalues of nested sequences of matrices resulting from random corank 1 projections of cl ..."
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Cited by 5 (1 self)
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Recent applications in queuing theory and statistical mechanics have isolated the process formed by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in general from the eigenvalues of nested sequences of matrices resulting from random corank 1 projections of classical random matrix ensembles, are identified for the LUE and JUE. The correlations for all these processes can be computed in a unified way. The resulting expressions can then be analyzed in various scaling limits. At the soft edge, with the rank of the minors differing by an amount proportional to N 2/3, the scaled correlations coincide with those known from the soft edge scaling of the Dyson Brownian motion model. 1 1
DETERMINANTAL TRANSITION KERNELS FOR SOME INTERACTING PARTICLES ON THE LINE
, 707
"... Abstract. We find the transition kernels for four Markovian interacting particle systems on the line, by proving that each of these kernels is intertwined with a KarlinMcGregor type kernel. The resulting kernels all inherit the determinantal structure from the KarlinMcGregor formula, and have a si ..."
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Cited by 5 (2 self)
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Abstract. We find the transition kernels for four Markovian interacting particle systems on the line, by proving that each of these kernels is intertwined with a KarlinMcGregor type kernel. The resulting kernels all inherit the determinantal structure from the KarlinMcGregor formula, and have a similar form to Schütz’s kernel for the totally asymmetric simple exclusion process. 1.
Slow decorrelations in KPZ growth
, 2008
"... For stochastic growth models in the KardarParisiZhang (KPZ) class in 1 + 1 dimensions, fluctuations grow as t 1/3 during time t and the correlation length at a fixed time scales as t 2/3. In this note we discuss the scale of time correlations. For a representant of the KPZ class, the polynuclear g ..."
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Cited by 4 (3 self)
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For stochastic growth models in the KardarParisiZhang (KPZ) class in 1 + 1 dimensions, fluctuations grow as t 1/3 during time t and the correlation length at a fixed time scales as t 2/3. In this note we discuss the scale of time correlations. For a representant of the KPZ class, the polynuclear growth model, we show that the spacetime is nontrivially fibred, having slow directions with decorrelation exponent equal to 1 instead of the usual 2/3. These directions are the characteristic curves of the PDE associated to the surface’s slope. As a consequence, previously proven results for spacelike paths will hold in the whole spacetime except along the slow curves. 1