# Height Fluctuations for the Stationary KPZ Equation

@article{Borodin2014HeightFF,
title={Height Fluctuations for the Stationary KPZ Equation},
author={Alexei Borodin and Ivan Corwin and Patrik L. Ferrari and B{\'a}lint Vető},
journal={Mathematical Physics, Analysis and Geometry},
year={2014},
volume={18},
pages={1-95}
}
• Published 25 July 2014
• Mathematics
• Mathematical Physics, Analysis and Geometry
We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X)$\mathcal {H}(0,X)=B(X)$, for B(X) a two-sided standard Brownian motion) and show that as time T goes to infinity, the fluctuations of the height function H(T,X)$\mathcal {H}(T,X)$ grow like T1/3 and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting…
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## References

SHOWING 1-10 OF 92 REFERENCES

### 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

### 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

### Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

• Mathematics
• 2005
The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0<ρ<1, is stationary in space and time. Let Nt(j) be the

### Crossover distributions at the edge of the rarefaction fan

• Mathematics
• 2010
We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_-<\rho_+$ so that macroscopically one has a

### Replica approach to the KPZ equation with the half Brownian motion initial condition

• Mathematics
• 2011
We consider the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We

• Mathematics
• 2014

### Fluctuation exponent of the KPZ/stochastic Burgers equation

• Mathematics, Physics
• 2011
(1.4) hε(t, x) = ε 1/2h(ε−zt, ε−1x). We will be considering these models in equilibrium, in which case h(t, x)−h(t, 0) is a two-sided Brownian motion with variance ν−1σ2 for each t. There are many

### The stochastic heat equation: Feynman-Kac formula and intermittence

• Mathematics, Physics
• 1995
We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a