Convergence towards Burger's equation and propagation of chaos for weakly asymmetric exclusion processes

Gärtner, Jürgen

doi:10.1016/0304-4149(87)90040-8

Cited by 89 publications

(110 citation statements)

References 3 publications

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“…Hence the integral in (5.11) is sensitive only to the right tail of g t (u). Furthermore, by the argument before (3.15), g t (u) tends to F TW (u), which has a right tail as exp[− 13) in agreement with (5.8), (5.9). As discussed in [18], Chapters 9 and 10, the connection to the δ-Bose gas was used to support the scaling exponent 1/3 and to study the right tail of the height distribution.…”

Section: (42)supporting

confidence: 79%

“…To exploit it, one notes that a particular exponential moment of the WASEP also satisfies a closed linear equation. This property was first proved by Gärtner [13,14], see also [9] and the related contribution [15]. For the PASEP with fixed p, q and arbitrary initial conditions, let us denote by E t the expectation with respect to the height statistics at time t and let us set f (j, t) = E t (e ϑh(j) ) .…”

Section: Centering and Exact Solutionmentioning

confidence: 92%

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Exact height distributions for the KPZ equation with narrow wedge initial condition

2010

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Abstract. We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x, t = 0) = −|x|/δ, δ 1. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution h(x, t) valid for any x and t > 0. The corresponding distribution function converges in the long time limit, t → ∞, to the Tracy-Widom distribution. The first order correction is a shift of order t −1/3 . We provide numerical computations based on the exact formula.arXiv:1002.1879v2 [cond-mat.stat-mech]

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Section: (42)supporting

confidence: 79%

Section: Centering and Exact Solutionmentioning

confidence: 92%

Exact height distributions for the KPZ equation with narrow wedge initial condition

2010

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“…Rather than directly studying the height function fluctuations, they looked at its Hopf-Cole transform (f → exp{−f }) which they called Z ǫ . Gärtner [74] had previously recognized that this transform linearizes the dynamics of the corner growth model into a discrete stochastic heat equation with multiplicative noise. Taking ǫ to zero, the solution (Z ǫ ) to the discrete stochastic heat equation converges to the solution (Z) to the continuum stochastic heat equation with multiplicative space-time white noise.…”

Section: Six Fundamental Initial Data Of the Kpz Equationmentioning

confidence: 99%

The Kardar–parisi–zhang Equation and Universality Class

Corwin

2012

Random Matrices: Theory Appl.

700

824

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Abstract. 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 Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -a non-linear 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 one-point 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.

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“…Following [9,12] we define in this section a microscopic Cole-Hopf transformation of the process η t . For N ≥ 1, let…”

Section: A Microscopic Cole-hopf Transformationmentioning

confidence: 99%

“…Note that we do not need to prove that this expression vanishes in the limit, as one would expect from the definition of the density fluctuation field, but just that it is uniformly bounded. The proof of the nonequilibrium density fluctuations we present here relies on a microscopic Cole-Hopf transformation introduced by Gärtner [12] to investigate the hydrodynamic behavior of weakly asymmetric exclusion processes on Z, and used by Dittrich and Gärtner [9] to prove the nonequilibrium fluctuations of the same models.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium fluctuations of one-dimensional boundary driven weakly asymmetric exclusion processes

Gonçalves¹,

Landim²,

Milanés³

2017

Ann. Appl. Probab.

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Convergence towards Burger's equation and propagation of chaos for weakly asymmetric exclusion processes

Cited by 89 publications

References 3 publications

Exact height distributions for the KPZ equation with narrow wedge initial condition

Exact height distributions for the KPZ equation with narrow wedge initial condition

The Kardar–parisi–zhang Equation and Universality Class

Nonequilibrium fluctuations of one-dimensional boundary driven weakly asymmetric exclusion processes

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