Stationary Correlations for the 1D KPZ Equation

@article{Imamura2013StationaryCF,
  title={Stationary Correlations for the 1D KPZ Equation},
  author={Takashi Imamura and Tomohiro Sasamoto},
  journal={Journal of Statistical Physics},
  year={2013},
  volume={150},
  pages={908-939}
}
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 initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit… 
View on Springer
arxiv.org
76 Citations
Highly Influential Citations
4
Background Citations
17
Methods Citations
16
Results Citations
2

76 Citations

Integration by Parts and the KPZ Two-Point Function
In this article we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to…
Large deviations for the KPZ equation from the KP equation
Recently, Quastel and Remenik \cite{QRKP} [arXiv:1908.10353] found a remarkable relation between some solutions of the finite time Kardar-Parisi-Zhang (KPZ) equation and the Kadomtsev-Petviashvili…
Height Fluctuations for the Stationary KPZ Equation
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…
Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case
We study the Kardar–Parisi–Zhang (KPZ) equation on the half-line x ⩾ 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on…
Coupled Kardar-Parisi-Zhang Equations in One Dimension
Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our…
Height distribution tails in the Kardar–Parisi–Zhang equation with Brownian initial conditions
For stationary interface growth, governed by the Kardar–Parisi–Zhang (KPZ) equation in 1+1 dimensions, typical fluctuations of the interface height at long times are described by the Baik–Rains…
Replica analysis of the one-dimensional KPZ equation
In the last few years several exact solutions have been obtained for the onedimensional KPZ equation, which describes the dynamics of growing interfaces. In particular the computations based on…
KP governs random growth off a 1-dimensional substrate
Abstract The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation.…
Simple derivation of the (– λ H)5/2 tail for the 1D KPZ equation
We study the long-time regime of the Kardar–Parisi–Zhang (KPZ) equation in 1  +  1 dimensions for the Brownian and droplet initial conditions and present a simple derivation of the tail of the large…
Reflected Brownian Motions in the KPZ Universality Class
A system of asymmetrically reflected Brownian motions is studied under various initial conditions. Asymmetric reflection means that each particle is reflected from its left neighbor. This system can…
...
...

References

SHOWING 1-10 OF 67 REFERENCES
Replica approach to the KPZ equation with the half Brownian motion initial condition
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…
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…
Exact solution for the stationary Kardar-Parisi-Zhang equation.
We obtain the first exact solution for the stationary one-dimensional Kardar-Parisi-Zhang equation. A formula for the distribution of the height is given in terms of a Fredholm determinant, which is…
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.
Exact Scaling Functions for One-Dimensional Stationary KPZ Growth
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…
One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality.
TLDR
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.
Fluctuation exponent of the KPZ/stochastic Burgers equation
(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…
Numerical evidence for stretched exponential relaxations in the Kardar-Parisi-Zhang equation.
  • E. Katzav, M. Schwartz
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2004
TLDR
This work presents results from extensive numerical integration of the Kardar-Parisi-Zhang equation in 1+1 dimensions aimed to check the long-time behavior of the dynamical structure factor of the KPZ system and gives an analytic expression that yields a very good approximation to the numerical data.
The one-dimensional KPZ equation and the Airy process
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…
Exact height distributions for the KPZ equation with narrow wedge initial condition
...
...