Two Speed TASEP

Borodin, Alexei; Ferrari, Patrik L.; Sasamoto, Tomohiro

doi:10.1007/s10955-009-9837-7

Cited by 48 publications

(67 citation statements)

References 39 publications

(86 reference statements)

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“…• One pt: F GUE [9,93] • Multi pt: Airy 2 [137,94,29] • One pt: F 0 [11,69] • Multi pt: Airy stat [10] • One pt: F GOE [12,13,70,146] • Multi pt: Airy 1 [31,32] • One pt: (F GOE ) 2 [11,8,136,19] • Multi pt: Airy 2→BM [90,46] • Multi pt: Airy 2→1 [33] • Multi pt: Airy 1→BM [34] KPZ/SHE…”

Section: Fluctuationsmentioning

confidence: 99%

The Kardar–parisi–zhang Equation and Universality Class

Corwin

2012

Random Matrices: Theory Appl.

703

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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Section: Fluctuationsmentioning

confidence: 99%

The Kardar–parisi–zhang Equation and Universality Class

Corwin

2012

Random Matrices: Theory Appl.

703

824

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“…x u ν , then one has the Lax-Oleinik variational formula for the solution, u = ∂ x h with (25) h(t, x) = sup…”

Section: Two Sided Brownian Motion B(x) X ∈ R Normalized To Have (19mentioning

confidence: 99%

Introduction to KPZ

Quastel

2011

Current Developments in Mathematics

177

140

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Abstract. This is an introductory survey of the Kardar-Parisi-Zhang equation (KPZ). The first chapter provides a non-rigorous background to the equation and to some of the many models which are supposed to lie in its universality class, as well as the predicted, non-standard fluctuations. The second chapter provides a rigorous introduction to the stochastic heat equation, whose logarithm is the solution of KPZ, as well as some of the known methods for proving convergence of discrete growth models and directed polymer free energies. Finally, we end with a sketch of the derivation of exact formulas for the one-point distributions of KPZ at finite time for special initial data, from the Tracy-Widom formulas for asymmetric exclusion.

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“…It has occurred also in the Corner Growth model [1], the PNG model [7], random matrices [2] and in TASEP [4,5,10]. In these earlier instances the transition was more or less indirectly related to an underlying determinantal structure.…”

Section: Theorem 2 Whenmentioning

confidence: 99%

On ASEP with Step Bernoulli Initial Condition

Tracy

Widom

2009

J Stat Phys

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This paper extends earlier work on ASEP to the case of step Bernoulli initial condition. The main results are a representation in terms of a Fredholm determinant for the probability distribution of a fixed particle, and asymptotic results which in particular establish KPZ universality for this probability in one regime. (And, as a corollary, for the current fluctuations.)Keywords Asymmetric simple exclusion process · KPZ universality · Bethe Ansatz

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Two Speed TASEP

Cited by 48 publications

References 39 publications

The Kardar–parisi–zhang Equation and Universality Class

The Kardar–parisi–zhang Equation and Universality Class

Introduction to KPZ

On ASEP with Step Bernoulli Initial Condition

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