Introduction to KPZ

@inproceedings{Quastel2011IntroductionTK,
  title={Introduction to KPZ},
  author={Jeremy Quastel},
  year={2011}
}
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… 

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TLDR
Examination of variants of the 3D DPRM, as well as numerically integrating, via the Itô prescription, the constrained SHE for different values of the KPZ coupling, provides strong evidence for universality within this 3D KPZ class, revealing shared values for the limit distribution skewness and kurtosis, along with universal first and second moments.

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Heat flows in 1+1 dimensional stochastic environment converge after scaling to the random geometry described by the directed landscape. In this first part, we show that the O'Connell-Yor polymer and
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References

SHOWING 1-10 OF 160 REFERENCES

Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions.

TLDR
This work provides the first exact calculation of the height distribution at arbitrary time t of the continuum Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with flat initial conditions and obtain the generating function of the moments of the directed polymer partition sum as a Fredholm Pfaffian.

Renormalization Fixed Point of the KPZ Universality Class

The one dimensional Kardar–Parisi–Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a

The Kardar-Parisi-Zhang Equation and Universality Class

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

Solving the KPZ equation

We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a

1 1 A pr 2 01 1 An exact solution for the KPZ equation with flat initial conditions

We provide the first exact calculation of the height distribution at arbitrary time t of the continuum KPZ growth equation in one dimension with flat initial conditions. We use the mapping onto a

On the long time behavior of the stochastic heat equation

Abstract We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum – the exact long time asymptotic. In the Carmona-Molchanov approach

A Fredholm Determinant Representation in ASEP

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer

21pYO-3 Spatial correlations of the 1D KPZ surface on a flat substrate

We study the spatial correlations of the one-dimensional KPZ surface for the flat initial condition. It is shown that the multi-point joint distribution for the height is given by a Fredholm
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