46 research outputs found
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
Slow decorrelations in KPZ growth
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 correlations. For a representant of the KPZ class, the polynuclear growth
model, we show that the space-time is non-trivially fibred, having slow
directions with decorrelation exponent equal to 1 instead of the usual 2/3.
These directions are the characteristic curves of the PDE associated to the
surface's slope. As a consequence, previously proven results for space-like
paths will hold in the whole space-time except along the slow curves.Comment: 22 pages, 9 figures, LaTeX; Minor language revision
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the
Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large
time distribution and processes and their dependence on the class on initial
condition. This means that the scaling exponents do not uniquely determine the
large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models.
Moreover, the limit process for curved limit shape turned out to show up in a
dynamical version of hermitian random matrices, but this analogy does not
extend to the case of symmetric matrices. Therefore the connections between
growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor
corrections in scaling of section 2.
Universal exit probabilities in the TASEP
We study the joint exit probabilities of particles in the totally asymmetric
simple exclusion process (TASEP) from space-time sets of given form. We extend
previous results on the space-time correlation functions of the TASEP, which
correspond to exits from the sets bounded by straight vertical or horizontal
lines. In particular, our approach allows us to remove ordering of time moments
used in previous studies so that only a natural space-like ordering of particle
coordinates remains. We consider sequences of general staircase-like boundaries
going from the northeast to southwest in the space-time plane. The exit
probabilities from the given sets are derived in the form of Fredholm
determinant defined on the boundaries of the sets. In the scaling limit, the
staircase-like boundaries are treated as approximations of continuous
differentiable curves. The exit probabilities with respect to points of these
curves belonging to arbitrary space-like path are shown to converge to the
universal Airy process.Comment: 46 pages, 7 figure
Statistics of layered zigzags: a two-dimensional generalization of TASEP
A novel discrete growth model in 2+1 dimensions is presented in three
equivalent formulations: i) directed motion of zigzags on a cylinder, ii)
interacting interlaced TASEP layers, and iii) growing heap over 2D substrate
with a restricted minimal local height gradient. We demonstrate that the
coarse-grained behavior of this model is described by the two-dimensional
Kardar-Parisi-Zhang equation. The coefficients of different terms in this
hydrodynamic equation can be derived from the steady state flow-density curve,
the so called `fundamental' diagram. A conjecture concerning the analytical
form of this flow-density curve is presented and is verified numerically.Comment: 5 pages, 4 figure
Exclusion type spatially heterogeneous processes in continuum
We study deterministic discrete time exclusion type spatially heterogeneous
particle processes in continuum. A typical example of this sort is a traffic
flow model with obstacles: traffic lights, speed bumps, spatially varying local
velocities etc. Ergodic averages of particle velocities are obtained and their
connections to other statistical quantities, in particular to particle and
obstacles densities (the so called Fundamental Diagram) is analyzed rigorously.
The main technical tool is a "dynamical" coupling construction applied in a
nonstandard fashion: instead of proving the existence of the successful
coupling (which even might not hold) we use its presence/absence as an
important diagnostic tool.Comment: 17 pages, minor corrections, accepted for publication in JSTA
On the partial connection between random matrices and interacting particle systems
In the last decade there has been increasing interest in the fields of random
matrices, interacting particle systems, stochastic growth models, and the
connections between these areas. For instance, several objects appearing in the
limit of large matrices arise also in the long time limit for interacting
particles and growth models. Examples of these are the famous Tracy-Widom
distribution functions and the Airy_2 process. The link is however sometimes
fragile. For example, the connection between the eigenvalues in the Gaussian
Orthogonal Ensembles (GOE) and growth on a flat substrate is restricted to
one-point distribution, and the connection breaks down if we consider the joint
distributions. In this paper we first discuss known relations between random
matrices and the asymmetric exclusion process (and a 2+1 dimensional
extension). Then, we show that the correlation functions of the eigenvalues of
the matrix minors for beta=2 Dyson's Brownian motion have, when restricted to
increasing times and decreasing matrix dimensions, the same correlation kernel
as in the 2+1 dimensional interacting particle system under diffusion scaling
limit. Finally, we analyze the analogous question for a diffusion on (complex)
sample covariance matrices.Comment: 31 pages, LaTeX; Added a section concerning the Markov property on
space-like path
Finite time corrections in KPZ growth models
We consider some models in the Kardar-Parisi-Zhang universality class, namely
the polynuclear growth model and the totally/partially asymmetric simple
exclusion process. For these models, in the limit of large time t, universality
of fluctuations has been previously obtained. In this paper we consider the
convergence to the limiting distributions and determine the (non-universal)
first order corrections, which turn out to be a non-random shift of order
t^{-1/3} (of order 1 in microscopic units). Subtracting this deterministic
correction, the convergence is then of order t^{-2/3}. We also determine the
strength of asymmetry in the exclusion process for which the shift is zero.
Finally, we discuss to what extend the discreteness of the model has an effect
on the fitting functions.Comment: 34 pages, 5 figures, LaTeX; Improved version including shift of PASEP
height functio
Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
The studies of fluctuations of the one-dimensional Kardar-Parisi-Zhang
universality class using the techniques from random matrix theory are reviewed
from the point of view of the asymmetric simple exclusion process. We explain
the basics of random matrix techniques, the connections to the polynuclear
growth models and a method using the Green's function.Comment: 41 pages, 10 figures, minor corrections, references adde
Airy processes and variational problems
We review the Airy processes; their formulation and how they are conjectured
to govern the large time, large distance spatial fluctuations of one
dimensional random growth models. We also describe formulas which express the
probabilities that they lie below a given curve as Fredholm determinants of
certain boundary value operators, and the several applications of these
formulas to variational problems involving Airy processes that arise in
physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI
Proceedings: Topics in percolative and disordered systems