157 research outputs found
Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks
Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system
come at two levels. On the central limit scale n^{1/2} one sees initial
fluctuations transported along characteristics and no dynamical noise. The
second order of fluctuations comes from the particle current across the
characteristic. For a system made up of independent random walks we show that
the second-order fluctuations appear at scale n^{1/4} and converge to a certain
self-similar Gaussian process. If the system is in equilibrium, this limiting
process specializes to fractional Brownian motion with Hurst parameter 1/4.
This contrasts with asymmetric exclusion and Hammersley's process whose
second-order fluctuations appear at scale n^{1/3}, as has been discovered
through related combinatorial growth models.Comment: Published at http://dx.doi.org/10.1214/009117904000000946 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Recent results and open problems on the hydrodynamics of disordered asymmetric exclusion and zero-range processes
This paper summarizes results and some open problems about the large-scale
and long-time behavior of asymmetric, disordered exclusion and zero-range
processes. These processes have randomly chosen jump rates at the sites of the
underlying lattice. The interesting feature is that for suitably distributed
random rates there is a phase transition where the process behaves differently
at high and low densities. Some of this distinction is visible on the
hydrodynamic scale.Comment: Proceedings of II Escola Brasileira de Probabilidade (Barra do Sahy,
Sao Paulo, August 1998
Strong law of large numbers for the interface in ballistic deposition
We prove a hydrodynamic limit for ballistic deposition on a multidimensional
lattice. In this growth model particles rain down at random and stick to the
growing cluster at the first point of contact. The theorem is that if the
initial random interface converges to a deterministic macroscopic function,
then at later times the height of the scaled interface converges to the
viscosity solution of a Hamilton-Jacobi equation. The proof idea is to
decompose the interface into the shapes that grow from individual seeds of the
initial interface. This decomposition converges to a variational formula that
defines viscosity solutions of the macrosopic equation. The technical side of
the proof involves subadditive methods and large deviation bounds for related
first-passage percolation processes
Fluctuation bounds for the asymmetric simple exclusion process
We give a partly new proof of the fluctuation bounds for the second class
particle and current in the stationary asymmetric simple exclusion process. One
novelty is a coupling that preserves the ordering of second class particles in
two systems that are themselves ordered coordinatewise.Comment: Minor improvements made to text. 24 page
A convexity property of expectations under exponential weights
Take a random variable X with some finite exponential moments. Define an
exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for
admissible values of the parameter t. Denote the weighted expectation of X
itself by r(t) = E^t(X), with inverse function t(r). We prove that for a convex
function f the expectation E^{t(r)}(f) is a convex function of the parameter r.
Along the way we develop correlation inequalities for convex functions.
Motivation for this result comes from equilibrium investigations of some
stochastic interacting systems with stationary product distributions. In
particular, convexity of the hydrodynamic flux function follows in some cases.Comment: After completion of this manuscript we learned that our main results
can be obtained as a special case of some propositions in Karlin: Total
Positivity, Vol.
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