# 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…

## 168 Citations

### A Riemann-Hilbert approach to the lower tail of the KPZ equation

- Mathematics
- 2019

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also…

### Local KPZ Behavior Under Arbitrary Scaling Limits

- MathematicsCommunications in Mathematical Physics
- 2022

A BSTRACT . One of the main difﬁculties in proving convergence of discrete models of surface growth to the Kardar–Parisi–Zhang (KPZ) equation in dimensions higher than one is that the correct way to…

### Time evolution of the Kardar-Parisi-Zhang equation

- Mathematics
- 2020

The use of the non-linear SPDEs are inevitable in both physics and applied mathematics since many of the physical phenomena in nature can be effectively modeled in random and non-linear way. The…

### The KPZ equation converges to the KPZ fixed point

- Mathematics
- 2020

We show that under the 1:2:3 scaling, critically probing large space and time, the solution of the KPZ equation starting from a continuous function plus a finite collection of narrow wedges converges…

### A combinatorial identity for the speed of growth in an anisotropic KPZ model

- Mathematics
- 2015

The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603-684], which belongs to the KPZ anisotropic universality class, was…

### Renormalization Fixed Point of the KPZ Universality Class

- Mathematics
- 2011

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…

### Aging for the stationary Kardar-Parisi-Zhang equation and related models

- Mathematics
- 2020

We study the aging property for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution to the stationary KPZ…

### Extremal paths, the stochastic heat equation, and the three-dimensional Kardar-Parisi-Zhang universality class.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2013

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.

### Singular HJB equations with applications to KPZ on the real line

- MathematicsProbability Theory and Related Fields
- 2022

This paper is devoted to studying the Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which is not well-defined in the classical sense and shall be understood by using…

### The heat and the landscape I

- Mathematics
- 2020

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…

## References

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

- Mathematics, PhysicsPhysical review letters
- 2011

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

- Mathematics
- 2011

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

- Mathematics
- 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…

### Solving the KPZ equation

- Mathematics
- 2011

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

- Mathematics, Physics
- 2013

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

- Mathematics
- 1999

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

- Mathematics
- 2008

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

- Mathematics
- 2005

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…