10 research outputs found
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 . 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
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
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 of a Hamilton-Jacobi
equation with random Hamiltonian
in dimension . It is known that homogenization occurs as , but little is known about the statistical fluctuations of .
Our main result shows that the variance of the solution is bounded
by . The proof relies on a modified Poincar\'e
inequality of Talagrand
A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
These notes are based on lectures delivered by the authors at a Langeoog
seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a
mixed audience of mathematicians and theoretical physicists. After a brief
outline of the basic physical concepts of equilibrium and nonequilibrium
states, the one-dimensional simple exclusion process is introduced as a
paradigmatic nonequilibrium interacting particle system. The stationary measure
on the ring is derived and the idea of the hydrodynamic limit is sketched. We
then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and
explain the associated universality conjecture for surface fluctuations in
growth models. This is followed by a detailed exposition of a seminal paper of
Johansson that relates the current fluctuations of the totally asymmetric
simple exclusion process (TASEP) to the Tracy-Widom distribution of random
matrix theory. The implications of this result are discussed within the
framework of the KPZ conjecture.Comment: 52 pages, 4 figures; to appear in J. Phys. A: Math. Theo
Occupation times of long-range exclusion and connections to KPZ class exponents
With respect to a class of long-range exclusion processes on \ZZ^d, with single particle transition rates of order , starting under Bernoulli invariant measure with density , we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on , and with respect to the variance of these functionals and associated scaling limits.
In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter .
However, in the asymmetric case, we study the asymptotics of the variances, which when and points to a curious dichotomy between long-range strength parameters . In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short-range model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.The research of CB was supported in part by the French Ministry of Education through the grant ANR JCJC EDNHS. PG thanks FCT (Portugal) for support through the research project PTDC/MAT/109844/2009 and CNPq (Brazil) for support through the research project 480431/2013-2. PG thanks CMAT for support by "FEDER" through the "Programa Operacional Factores de Competitividade COMPETE" and by FCT through the project PEst-C/MAT/UI0013/2011. SS was supported in part by ARO grant W911NF-14-1-0179
Stationary cocycles and Busemann functions for the corner growth model
We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, out- side of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as bound- ary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles are used to prove results about semi-infinite geodesics and the competition interface