S. Chhita, P.L. Ferrari and H. Spohn Limit distributions for KPZ growth models with spatially homogeneous random initial conditions preprint, arXiv:1611.06690 2016 http://arxiv.org/abs/1611.06690
Abstract: For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the roughness of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six-vertex model at is conical point.
P.L. Ferrari and H. Spohn On time correlations for KPZ growth in one dimension preprint: arXiv:1602.00486 2016 http://arxiv.org/abs/1602.00486
Abstract: 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 each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the large time behavior for the covariance of the height gradients.
2015
Patrik L. Ferrari, Herbert Spohn and Thomas Weiss Scaling Limit for Brownian Motions with One-sided Collisions Ann. Appl. Probab., 25(3): 1349-1382 2015 http://dx.doi.org/10.1214/14-AAP1025
Abstract: We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, i.e., particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the Airy\(_1\) process.
Patrik L. Ferrari, Herbert Spohn and Thomas Weiss Brownian motions with one-sided collisions: the stationary case Electronic Journal of Probability, 20(Art. 69): 1-41 2015 http://dx.doi.org/10.1214/EJP.v20-4177
Abstract: We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the large time limit. This leads to a new universal cross-over process.
2014
Gerard Barkema, Patrik L. Ferrari, Joel L. Lebowitz and Herbert Spohn KPZ universality class and the anchored Toom interface Phys. Rev. E, 90(Art. 042116) 2014 http://dx.doi.org/10.1103/PhysRevE.90.042116
Abstract: We revisit the anchored Toom interface and use KPZ scaling theory to argue that the interface fluctuations are governed by the Airy1 process with the role of space and time interchanged. There is no free parameter. The predictions are numerically well confirmed for space-time statistics in the stationary state. In particular the spatial fluctuations of the interface are given by the GOE edge distribution of Tracy and Widom.
2013
Patrik L. Ferrari, Tomohiro Sasamoto and Herbert Spohn Coupled Kardar-Parisi-Zhang Equations in One Dimension J. Stat. Phys., 153(3): 377-399 2013 http://dx.doi.org/10.1007/s10955-013-0842-5
Abstract: 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 contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.