Related papers: Spatial decorrelation of KPZ from narrow wedge
We study the fixed-time spatial covariance of the KPZ equation with flat initial profile. Using Malliavin calculus and a Clark-Ocone representation, we show that as $|x|\to\infty$, $\mathrm{Cov}[h(t,x),h(t,0)]$ is governed by a…
We consider the KPZ equation in one space dimension with narrow wedge initial condition, $h(x,t=0)=- |x|/\delta$, $\delta\ll 1$. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we…
Let H(t,x) be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac…
We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equation started from the narrow wedge initial condition. In this article, we ask how the peaks and valleys of the KPZ height function (centered by time/24) at any spatial…
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 determinant, with its kernel in the scaling limit…
For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in 1+1 dimensions, fluctuations grow as t^{1/3} during time t and the correlation length at a fixed time scales as t^{2/3}. In this note we discuss the scale of time…
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…
The early time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension, starting from a Brownian initial condition with a drift $w$, is studied using the exact Fredholm determinant representation. For large drift we recover the…
This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces.
Consider a discrete one-dimensional random surface whose height at a point grows as a function of the heights at neighboring points plus an independent random noise. Assuming that this function is equivariant under constant shifts,…
We consider the narrow wedge solution to the Kardar-Parisi-Zhang stochastic PDE under the characteristic $3:2:1$ scaling of time, space and fluctuations. We study the correlation of fluctuations at two different times. We show that when the…
We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data 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…
Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In…
There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting…
We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the…
We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field, stationary and white-noise in time but fractional Brownian in space with roughness exponent…
We study the solution of the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height $h(x,t)$ on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a…
We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as…
We consider the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We employ the replica Bethe ansatz and show that the…
The time-dependent probability distribution function of the height for the Kardar-Parisi-Zhang equation with sharp wedge initial conditions has been obtained recently as a convolution between the Gumbel distribution and a difference of two…