Microscopic concavity and fluctuation bounds in a class of deposition processes

We prove fluctuation bounds for the particle current in totally asymmetric zero range processes in one dimension with nondecreasing, concave jump rates whose slope decays exponentially. Fluctuations in the characteristic directions have order of magnitude $t^{1/3}$. This is in agreement with the expectation that these systems lie in the same KPZ universality class as the asymmetric simple exclusion process. The result is via a robust argument formulated for a broad class of deposition-type processes. Besides this class of zero range processes, hypotheses of this argument have also been verified in the authors' earlier papers for the asymmetric simple exclusion and the constant rate zero range processes, and are currently under development for a bricklayers process with exponentially increasing jump rates.Comment: Improved after Referee's comments: we added explanations and changed some parts of the text. 50 pages, 1 figur

There are No Nice Interfaces in 2+1 Dimensional SOS-Models in Random Media

We prove that in dimension $d\leq 2$ translation covariant Gibbs states describing rigid interfaces in a disordered solid-on-solid (SOS) cannot exist for any value of the temperature, in contrast to the situation in $d\geq 3$. The prove relies on an adaptation of a theorem of Aizenman and Wehr. Keywords: Disordered systems, interfaces, SOS-modelComment: 8 pages, gz-compressed Postscrip

Selected recollections of my relationship with Leo Breiman

During the period 1962--1964, I had a tenure track Assistant Professorship in Mathematics at Cornell University in Ithaca, New York, where I did research in probability theory, especially on linear diffusion processes. Being somewhat lonely there and not liking the cold winter weather, I decided around the beginning of 1964 to try to get a job in the Mathematics Department at UCLA, in the city in which I was born and raised. At that time, Leo Breiman was an Associate Professor in that department. Presumably, he liked my research on linear diffusion processes and other research as well, since the department offered me a tenure track Assistant Professorship, which I happily accepted. During the Summer of 1965, I worked on various projects with Sidney Port, then at RAND Corporation, especially on random walks and related material. I was promoted to Associate Professor, effective in Fall, 1966, presumably thanks in part to Leo. Early in 1966, I~was surprised to be asked by Leo to participate in a department meeting called to discuss the possible hiring of Sidney. The conclusion was that Sidney was hired as Associate Professor in the department, as of Fall, 1966. Leo communicated to me his view that he thought that Sidney and I worked well together, which is why he had urged the department to hire Sidney. Anyhow, Sidney and I had a very fruitful and enjoyable collaboration in probability and, to a much lesser extent, in theoretical statistics, for a number of years thereafter.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS431 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in TASEP with the step-type initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure

A multi-layer extension of the stochastic heat equation

Motivated by recent developments on solvable directed polymer models, we define a 'multi-layer' extension of the stochastic heat equation involving non-intersecting Brownian motions.Comment: v4: substantially extended and revised versio

Locating the minimum : Approach to equilibrium in a disordered, symmetric zero range process

We consider the dynamics of the disordered, one-dimensional, symmetric zero range process in which a particle from an occupied site $k$ hops to its nearest neighbour with a quenched rate $w(k)$. These rates are chosen randomly from the probability distribution $f(w) \sim (w-c)^{n}$, where $c$ is the lower cutoff. For $n > 0$, this model is known to exhibit a phase transition in the steady state from a low density phase with a finite number of particles at each site to a high density aggregate phase in which the site with the lowest hopping rate supports an infinite number of particles. In the latter case, it is interesting to ask how the system locates the site with globally minimum rate. We use an argument based on local equilibrium, supported by Monte Carlo simulations, to describe the approach to the steady state. We find that at large enough time, the mass transport in the regions with a smooth density profile is described by a diffusion equation with site-dependent rates, while the isolated points where the mass distribution is singular act as the boundaries of these regions. Our argument implies that the relaxation time scales with the system size $L$ as $L^{z}$ with $z=2+1/(n+1)$ for $n > 1$ and suggests a different behaviour for $n < 1$.Comment: Revtex, 7 pages including 3 figures. Submitted to Pramana -- special issue on mesoscopic and disordered system

A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation

We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^\epsilon$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\omega) = K(p) - V(x/\epsilon,\omega)$ in dimension $d \geq 2$. It is known that homogenization occurs as $\epsilon \to 0$, but little is known about the statistical fluctuations of $w^\epsilon$. Our main result shows that the variance of the solution $w^\epsilon$ is bounded by $O(\epsilon/|\log \epsilon|)$. The proof relies on a modified Poincar\'e inequality of Talagrand

Occupation times of exclusion processes

In this paper we consider exclusion processes $\{\eta_t: t\geq{0}\}$ evolving on the one-dimensional lattice $\mathbb{Z}$, under the diffusive time scale $tn^2$ and starting from the invariant state $\nu_\rho$ - the Bernoulli product measure of parameter $\rho\in{[0,1]}$. Our goal consists in establishing the scaling limits of the additive functional $\Gamma_t:=\int_{0}^{tn^2} \eta_s(0)\, ds$ - {\em{ the occupation time of the origin}}. We present a method, recently introduced in \cite{G.J.}, from which a {\em{local Boltzmann-Gibbs Principle}} can be derived for a general class of exclusion processes. In this case, this principle says that $\Gamma_t$ is very well approximated to the additive functional of the density of particles. As a consequence, the scaling limits of $\Gamma_t$ follow from the scaling limits of the density of particles. As examples we present the mean-zero exclusion, the symmetric simple exclusion and the weakly asymmetric simple exclusion. For the latter under a strong asymmetry regime, the limit of $\Gamma_t$ is given in terms of the solution of the KPZ equation.FC

Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure

In [arXiv:0804.3035] we studied an interacting particle system which can be also interpreted as a stochastic growth model. This model belongs to the anisotropic KPZ class in 2+1 dimensions. In this paper we present the results that are relevant from the perspective of stochastic growth models, in particular: (a) the surface fluctuations are asymptotically Gaussian on a sqrt(ln(t)) scale and (b) the correlation structure of the surface is asymptotically given by the massless field.Comment: 13 pages, 4 figure

Generalized Green Functions and current correlations in the TASEP

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinantal formula for the generalized Green function which describes transitions between positions of particles at different individual time moments. In particular, the generalized Green function defines a probability measure at staircase lines on the space-time plane. The marginals of this measure are the TASEP correlation functions in the space-time region not covered by the standard Green function approach. As an example, we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance. An asymptotic analysis shows that current fluctuations converge to the ${Airy}_2$ process.Comment: 46 pages, 3 figure