181 research outputs found
Statistics of Multiple Sign Changes in a Discrete Non-Markovian Sequence
We study analytically the statistics of multiple sign changes in a discrete
non-Markovian sequence ,\psi_i=\phi_i+\phi_{i-1} (i=1,2....,n) where \phi_i's
are independent and identically distributed random variables each drawn from a
symmetric and continuous distribution \rho(\phi). We show that the probability
P_m(n) of m sign changes upto n steps is universal, i.e., independent of the
distribution \rho(\phi). The mean and variance of the number of sign changes
are computed exactly for all n>0. We show that the generating function {\tilde
P}(p,n)=\sum_{m=0}^{\infty}P_m(n)p^m\sim \exp[-\theta_d(p)n] for large n where
the `discrete' partial survival exponent \theta_d(p) is given by a nontrivial
formula, \theta_d(p)=\log[{{\sin}^{-1}(\sqrt{1-p^2})}/{\sqrt{1-p^2}}] for 0\le
p\le 1. We also show that in the natural scaling limit when m is large, n is
large but but keeping x=m/n fixed, P_m(n)\sim \exp[-n \Phi(x)] where the large
deviation function \Phi(x) is computed. The implications of these results for
Ising spin glasses are discussed.Comment: 4 pages revtex, 1 eps figur
Persistence in higher dimensions : a finite size scaling study
We show that the persistence probability , in a coarsening system of
linear size at a time , has the finite size scaling form where is the persistence exponent and
is the coarsening exponent. The scaling function for
and is constant for large . The scaling form implies a fractal
distribution of persistent sites with power-law spatial correlations. We study
the scaling numerically for Glauber-Ising model at dimension to 4 and
extend the study to the diffusion problem. Our finite size scaling ansatz is
satisfied in all these cases providing a good estimate of the exponent
.Comment: 4 pages in RevTeX with 6 figures. To appear in Phys. Rev.
Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models
We compute exactly the distribution of the occupation time in a discrete {\em
non-Markovian} toy sequence which appears in various physical contexts such as
the diffusion processes and Ising spin glass chains. The non-Markovian property
makes the results nontrivial even for this toy sequence. The distribution is
shown to have non-Gaussian tails characterized by a nontrivial large deviation
function which is computed explicitly. An exact mapping of this sequence to an
Ising spin glass chain via a gauge transformation raises an interesting new
question for a generic finite sized spin glass model: at a given temperature,
what is the distribution (over disorder) of the thermally averaged number of
spins that are aligned to their local fields? We show that this distribution
remains nontrivial even at infinite temperature and can be computed explicitly
in few cases such as in the Sherrington-Kirkpatrick model with Gaussian
disorder.Comment: 10 pages Revtex (two-column), 1 eps figure (included
Exact Phase Diagram of a model with Aggregation and Chipping
We revisit a simple lattice model of aggregation in which masses diffuse and
coalesce upon contact with rate 1 and every nonzero mass chips off a single
unit of mass to a randomly chosen neighbour with rate . The dynamics
conserves the average mass density and in the stationary state the
system undergoes a nonequilibrium phase transition in the plane
across a critical line . In this paper, we show analytically that in
arbitrary spatial dimensions, exactly and hence,
remarkably, independent of dimension. We also provide direct and indirect
numerical evidence that strongly suggest that the mean field asymptotic answer
for the single site mass distribution function and the associated critical
exponents are super-universal, i.e., independent of dimension.Comment: 11 pages, RevTex, 3 figure
Persistence in a Stationary Time-series
We study the persistence in a class of continuous stochastic processes that
are stationary only under integer shifts of time. We show that under certain
conditions, the persistence of such a continuous process reduces to the
persistence of a corresponding discrete sequence obtained from the measurement
of the process only at integer times. We then construct a specific sequence for
which the persistence can be computed even though the sequence is
non-Markovian. We show that this may be considered as a limiting case of
persistence in the diffusion process on a hierarchical lattice.Comment: 8 pages revte
Persistence of a Continuous Stochastic Process with Discrete-Time Sampling: Non-Markov Processes
We consider the problem of `discrete-time persistence', which deals with the
zero-crossings of a continuous stochastic process, X(T), measured at discrete
times, T = n(\Delta T). For a Gaussian Stationary Process the persistence (no
crossing) probability decays as exp(-\theta_D T) = [\rho(a)]^n for large n,
where a = \exp[-(\Delta T)/2], and the discrete persistence exponent, \theta_D,
is given by \theta_D = \ln(\rho)/2\ln(a). Using the `Independent Interval
Approximation', we show how \theta_D varies with (\Delta T) for small (\Delta
T) and conclude that experimental measurements of persistence for smooth
processes, such as diffusion, are less sensitive to the effects of discrete
sampling than measurements of a randomly accelerated particle or random walker.
We extend the matrix method developed by us previously [Phys. Rev. E 64,
015151(R) (2001)] to determine \rho(a) for a two-dimensional random walk and
the one-dimensional random acceleration problem. We also consider `alternating
persistence', which corresponds to a < 0, and calculate \rho(a) for this case.Comment: 14 pages plus 8 figure
Nonequilibrium phase transitions in models of adsorption and desorption
The nonequilibrium phase transition in a system of diffusing, coagulating
particles in the presence of a steady input and evaporation of particles is
studied. The system undergoes a transition from a phase in which the average
number of particles is finite to one in which it grows linearly in time. The
exponents characterizing the mass distribution near the critical point are
calculated in all dimensions.Comment: 10 pages, 2 figures (To appear in Phys. Rev. E
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting
Scaling fields in the two-dimensional abelian sandpile model
We consider the isotropic two-dimensional abelian sandpile model from a
perspective based on two-dimensional (conformal) field theory. We compute
lattice correlation functions for various cluster variables (at and off
criticality), from which we infer the field-theoretic description in the
scaling limit. We find a perfect agreement with the predictions of a c=-2
conformal field theory and its massive perturbation, thereby providing direct
evidence for conformal invariance and more generally for a description in terms
of a local field theory. The question of the height 2 variable is also
addressed, with however no definite conclusion yet.Comment: 22 pages, 1 figure (eps), uses revte
Avalanches and the Renormalization Group for Pinned Charge-Density Waves
The critical behavior of charge-density waves (CDWs) in the pinned phase is
studied for applied fields increasing toward the threshold field, using
recently developed renormalization group techniques and simulations of
automaton models. Despite the existence of many metastable states in the pinned
state of the CDW, the renormalization group treatment can be used successfully
to find the divergences in the polarization and the correlation length, and, to
first order in an expansion, the diverging time scale. The
automaton models studied are a charge-density wave model and a ``sandpile''
model with periodic boundary conditions; these models are found to have the
same critical behavior, associated with diverging avalanche sizes. The
numerical results for the polarization and the diverging length and time scales
in dimensions are in agreement with the analytical treatment. These
results clarify the connections between the behaviour above and below
threshold: the characteristic correlation lengths on both sides of the
transition diverge with different exponents. The scaling of the distribution of
avalanches on the approach to threshold is found to be different for automaton
and continuous-variable models.Comment: 29 pages, 11 postscript figures included, REVTEX v3.0 (dvi and PS
files also available by anonymous ftp from external.nj.nec.com in directory
/pub/alan/cdwfigs
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