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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 97 (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.
Free energy fluctuations for directed polymers in random media in 1+1 dimension
, 2012
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Fluctuation exponent of the KPZ/stochastic Burgers equation
 J. Amer. Math. Soc
"... 1.1. Background. The KardarParisiZhang (KPZ) equation [14] is a formal stochastic partial differential equation for a random function h(t, x), t>0, x ∈ R, (1.1) ∂th = −λ(∂xh) 2 + ν ∂ 2 xh + σ ˙ W, where ν>0andσ, λ = 0 are fixed parameters and ˙ W (t, x) is Gaussian spacetime ..."
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Cited by 11 (1 self)
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1.1. Background. The KardarParisiZhang (KPZ) equation [14] is a formal stochastic partial differential equation for a random function h(t, x), t>0, x ∈ R, (1.1) ∂th = −λ(∂xh) 2 + ν ∂ 2 xh + σ ˙ W, where ν>0andσ, λ = 0 are fixed parameters and ˙ W (t, x) is Gaussian spacetime
Transition probabilities of the Bethe ansatz solvable interacting particle systems
 J. Stat. Phys
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4 GLOBAL ASYMPTOTICS FOR THE CHRISTOFFELDARBOUX KERNEL OF RANDOM MATRIX THEORY
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Fluctuations for TASEP and LPP with general initial data
 In preparation
"... We prove Airy process variational formulas for the onepoint probability distribution of (discrete time parallel update) TASEP with general initial data, as well as last passage percolation from a general downright lattice path to a point. We also consider variants of last passage percolation with ..."
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Cited by 2 (2 self)
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We prove Airy process variational formulas for the onepoint probability distribution of (discrete time parallel update) TASEP with general initial data, as well as last passage percolation from a general downright lattice path to a point. We also consider variants of last passage percolation with inhomogeneous parameter geometric weights and provide variational formulas of a similar nature. This proves one aspect of the conjectural description of the renormalization fixed point of the KardarParisiZhang universality class. 1