27,477 research outputs found
Numerical analysis of the master equation
Applied to the master equation, the usual numerical integration methods, such
as Runge-Kutta, become inefficient when the rates associated with various
transitions differ by several orders of magnitude. We introduce an integration
scheme that remains stable with much larger time increments than can be used in
standard methods. When only the stationary distribution is required, a direct
iteration method is even more rapid; this method may be extended to construct
the quasi-stationary distribution of a process with an absorbing state.
Applications to birth-and-death processes reveal gains in efficiency of two or
more orders of magnitude.Comment: 7 pages 3 figure
On the asymmetric zero-range in the rarefaction fan
We consider the one-dimensional asymmetric zero-range process starting from a
step decreasing profile. In the hydrodynamic limit this initial condition leads
to the rarefaction fan of the associated hydrodynamic equation. Under this
initial condition and for totally asymmetric jumps, we show that the weighted
sum of joint probabilities for second class particles sharing the same site is
convergent and we compute its limit. For partially asymmetric jumps we derive
the Law of Large Numbers for the position of a second class particle under the
initial configuration in which all the positive sites are empty, all the
negative sites are occupied with infinitely many first class particles and with
a single second class particle at the origin. Moreover, we prove that among the
infinite characteristics emanating from the position of the second class
particle, this particle chooses randomly one of them. The randomness is given
in terms of the weak solution of the hydrodynamic equation through some sort of
renormalization function. By coupling the zero-range with the exclusion process
we derive some limiting laws for more general initial conditions.Comment: 22 pages, to appear in Journal of Statistical Physic
No phase transition for Gaussian fields with bounded spins
Let a<b, \Omega=[a,b]^{\Z^d} and H be the (formal) Hamiltonian defined on
\Omega by
H(\eta) = \frac12 \sum_{x,y\in\Z^d} J(x-y) (\eta(x)-\eta(y))^2 where
J:\Z^d\to\R is any summable non-negative symmetric function (J(x)\ge 0 for all
x\in\Z^d, \sum_x J(x)<\infty and J(x)=J(-x)). We prove that there is a unique
Gibbs measure on \Omega associated to H. The result is a consequence of the
fact that the corresponding Gibbs sampler is attractive and has a unique
invariant measure.Comment: 7 page
Inflated Beta Distributions
This paper considers the issue of modeling fractional data observed in the
interval [0,1), (0,1] or [0,1]. Mixed continuous-discrete distributions are
proposed. The beta distribution is used to describe the continuous component of
the model since its density can have quite diferent shapes depending on the
values of the two parameters that index the distribution. Properties of the
proposed distributions are examined. Also, maximum likelihood and method of
moments estimation is discussed. Finally, practical applications that employ
real data are presented.Comment: 15 pages, 4 figures. Submitted to Statistical Paper
Competition interfaces and second class particles
The one-dimensional nearest-neighbor totally asymmetric simple exclusion
process can be constructed in the same space as a last-passage percolation
model in Z^2. We show that the trajectory of a second class particle in the
exclusion process can be linearly mapped into the competition interface between
two growing clusters in the last-passage percolation model. Using technology
built up for geodesics in percolation, we show that the competition interface
converges almost surely to an asymptotic random direction. As a consequence we
get a new proof for the strong law of large numbers for the second class
particle in the rarefaction fan and describe the distribution of the asymptotic
angle of the competition interface.Comment: Published at http://dx.doi.org/10.1214/009117905000000080 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Universality of slow decorrelation in KPZ growth
There has been much success in describing the limiting spatial fluctuations
of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper
rescaling of time should introduce a non-trivial temporal dimension to these
limiting fluctuations. In one-dimension, the KPZ class has the dynamical
scaling exponent , that means one should find a universal space-time
limiting process under the scaling of time as , space like
and fluctuations like as .
In this paper we provide evidence for this belief. We prove that under
certain hypotheses, growth models display temporal slow decorrelation. That is
to say that in the scalings above, the limiting spatial process for times and are identical, for any . The hypotheses are known
to be satisfied for certain last passage percolation models, the polynuclear
growth model, and the totally / partially asymmetric simple exclusion process.
Using slow decorrelation we may extend known fluctuation limit results to
space-time regions where correlation functions are unknown.
The approach we develop requires the minimal expected hypotheses for slow
decorrelation to hold and provides a simple and intuitive proof which applied
to a wide variety of models.Comment: Exposition improved, typos correcte
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