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

@article{Calabrese2011ExactSF, title={Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions.}, author={Pasquale Calabrese and Pierre Le Doussal}, journal={Physical review letters}, year={2011}, volume={106 25}, pages={ 250603 } }

We provide 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. We use the mapping onto a directed polymer with one end fixed, one free, and the Bethe ansatz for the replicated attractive boson model. We obtain the generating function of the moments of the directed polymer partition sum as a Fredholm Pfaffian. Our formula, valid for all times, exhibits convergence of…

## 174 Citations

### 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…

### Height distribution of the Kardar-Parisi-Zhang equation with sharp-wedge initial condition: numerical evaluations.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2011

This work evaluates numerically this distribution of the height for the Kardar-Parisi-Zhang equation with sharp wedge initial conditions as a convolution between the Gumbel distribution and a difference of two Fredholm determinants over the whole time span.

### Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall.

- MathematicsPhysical review. E
- 2021

We present an exact solution for the height distribution of the KPZ equation at any time t in a half space with flat initial condition. This is equivalent to obtaining the free-energy distribution of…

### Probing large deviations of the Kardar-Parisi-Zhang equation at short times with an importance sampling of directed polymers in random media.

- PhysicsPhysical review. E
- 2020

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature, showing spectacular agreement with the analytical expressions.

### Introduction to KPZ

- Mathematics
- 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…

### Universal fluctuations in Kardar-Parisi-Zhang growth on one-dimensional flat substrates.

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

A numerical study of the evolution of height distributions obtained in interface growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class yields evidences for the universality of the GOE distribution in KPZ growth on flat substrates.

### 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.

### 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…

### Stationary Correlations for the 1D KPZ Equation

- Mathematics
- 2013

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the…

### Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions

- MathematicsNuclear Physics B
- 2018

## References

SHOWING 1-10 OF 58 REFERENCES

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

- Mathematics, PhysicsPhysical review letters
- 2010

The solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force, and provides a determinantal formula for the probability distribution function of the height h(x,t) for all t>0.

### Exact Scaling Functions for One-Dimensional Stationary KPZ Growth

- Mathematics, Physics
- 2004

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence…

### Variational Formulation for the Kpz and Related Kinetic Equations

- MathematicsInt. J. Bifurc. Chaos
- 2009

A variational formulation for the Kardar–Parisi–Zhang equation is presented that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations, and some global shift invariance properties previously conjectured are proved.

### The one-dimensional KPZ equation and the Airy process

- Mathematics
- 2011

Our previous work on the one-dimensional KPZ equation with sharp wedge initial data is extended to the case of the joint height statistics at n spatial points for some common fixed time. Assuming a…

### Replica Bethe ansatz derivation of the Tracy–Widom distribution of the free energy fluctuations in one-dimensional directed polymers

- Physics
- 2010

The distribution function of the free energy fluctuations in one-dimensional directed polymers with a δ-correlated random potential is studied by mapping the replicated problem to the N-particle…

### Free-energy distribution of the directed polymer at high temperature

- Mathematics, Physics
- 2010

We study the directed polymer of length t in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal…

### Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy-Widom distribution.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2004

We compute exactly the asymptotic distribution of scaled height in a (1+1)-dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing…

### Bethe ansatz solution for one-dimensional directed polymers in random media

- Physics
- 2010

We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many-body quantum boson system with attractive…

### Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues

- Mathematics
- 2004

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the…