# One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality.

@article{Sasamoto2010OnedimensionalKE, title={One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality.}, author={Tomohiro Sasamoto and Herbert Spohn}, journal={Physical review letters}, year={2010}, volume={104 23}, pages={ 230602 } }

We report on the first exact solution of the Kardar-Parisi-Zhang (KPZ) equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile. The solution provides a determinantal formula for the probability distribution function of the height h(x,t) for all t>0. In particular, we show that for large t, on the scale t(1/3), the statistics is given by the Tracy-Widom distribution, known already from the Gaussian unitary…

## 302 Citations

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.

The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

- Physics
- 2019

The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation…

One-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: Limit distributions.

- MathematicsPhysical review. E
- 2020

It is established that the statistical properties of the one-dimensional (1D) KS PDE in this state are in the 1D KPZ universality class.

Short-time growth of a Kardar-Parisi-Zhang interface with flat initial conditions.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

The short-time behavior of the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) growth equation with a flat initial condition is obtained from the exact expressions for the moments of the partition…

The 1D Kardar–Parisi–Zhang equation: Height distribution and universality

- Physics
- 2016

The Kardar–Parisi–Zhang (KPZ) equation, which was introduced in 1986 as a model equation to describe the dynamics of an interface motion, has been attracting renewed interest in recent years. In…

Non-KPZ fluctuations in the derivative of the Kardar-Parisi-Zhang equation or noisy Burgers equation.

- Mathematics, PhysicsPhysical review. E
- 2020

It is shown that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the field statistics behave in a remarkably different way: while the KPZ equation follows the Tracy-Widom distribution, its derivative displays Gaussian behavior, hence being in a different universality class.

Competing Universalities in Kardar-Parisi-Zhang Growth Models.

- MathematicsPhysical review letters
- 2019

A phenomenological theory is proposed to explain the universality of height fluctuations at the crossing point of two interacting (1+1)-dimensional Kardar-Parisi-Zhang interfaces with curved and flat initial conditions and discuss possible applications in nonequilibrium transport and traffic flow.

Dynamics of surface roughening in the space-fractional Kardar–Parisi–Zhang growth: numerical results

- Physics, Mathematics
- 2012

We numerically study the (1+1)-dimensional space-fractional Kardar–Parisi–Zhang (SFKPZ) equation describing surface roughening in the presence of anomalous diffusion based on the Riesz-type…

Coulomb-Gas Electrostatics Controls Large Fluctuations of the Kardar-Parisi-Zhang Equation.

- MathematicsPhysical review letters
- 2018

A large deviation principle is established for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition and rigorous proof of finite-time tail bounds on the KPZ distribution is provided.

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