99 research outputs found
On the distribution of surface extrema in several one- and two-dimensional random landscapes
We study here a standard next-nearest-neighbor (NNN) model of ballistic
growth on one- and two-dimensional substrates focusing our analysis on the
probability distribution function of the number of maximal points
(i.e., local ``peaks'') of growing surfaces. Our analysis is based on two
central results: (i) the proof (presented here) of the fact that uniform
one--dimensional ballistic growth process in the steady state can be mapped
onto ''rise-and-descent'' sequences in the ensemble of random permutation
matrices; and (ii) the fact, established in Ref. \cite{ov}, that different
characteristics of ``rise-and-descent'' patterns in random permutations can be
interpreted in terms of a certain continuous--space Hammersley--type process.
For one--dimensional system we compute exactly and also present
explicit results for the correlation function characterizing the enveloping
surface. For surfaces grown on 2d substrates, we pursue similar approach
considering the ensemble of permutation matrices with long--ranged
correlations. Determining exactly the first three cumulants of the
corresponding distribution function, we define it in the scaling limit using an
expansion in the Edgeworth series, and show that it converges to a Gaussian
function as .Comment: 25 pages, 12 figure
Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results
We analyze the mean time t_{app} that a randomly moving particle spends in a
bounded domain (sphere) before it escapes through a small window in the
domain's boundary. A particle is assumed to diffuse freely in the bulk until it
approaches the surface of the domain where it becomes weakly adsorbed, and then
wanders diffusively along the boundary for a random time until it desorbs back
to the bulk, and etc. Using a mean-field approximation, we define t_{app}
analytically as a function of the bulk and surface diffusion coefficients, the
mean time it spends in the bulk between two consecutive arrivals to the surface
and the mean time it wanders on the surface within a single round of the
surface diffusion.Comment: 8 pages, 1 figure, submitted to JC
Random patterns generated by random permutations of natural numbers
We survey recent results on some one- and two-dimensional patterns generated
by random permutations of natural numbers. In the first part, we discuss
properties of random walks, evolving on a one-dimensional regular lattice in
discrete time , whose moves to the right or to the left are induced by the
rise-and-descent sequence associated with a given random permutation. We
determine exactly the probability of finding the trajectory of such a
permutation-generated random walk at site at time , obtain the
probability measure of different excursions and define the asymptotic
distribution of the number of "U-turns" of the trajectories - permutation
"peaks" and "through". In the second part, we focus on some statistical
properties of surfaces obtained by randomly placing natural numbers on sites of a 1d or 2d square lattices containing sites. We
calculate the distribution function of the number of local "peaks" - sites the
number at which is larger than the numbers appearing at nearest-neighboring
sites - and discuss some surprising collective behavior emerging in this model.Comment: 16 pages, 5 figures; submitted to European Physical Journal,
proceedings of the conference "Stochastic and Complex Systems: New Trends and
Expectations" Santander, Spai
Diffusion and subdiffusion of interacting particles on comb-like structures
We study the dynamics of a tracer particle (TP) on a comb lattice populated
by randomly moving hard-core particles in the dense limit. We first consider
the case where the TP is constrained to move on the backbone of the comb only,
and, in the limit of high density of particles, we present exact analytical
results for the cumulants of the TP position, showing a subdiffusive behavior
. At longer times, a second regime is observed, where standard
diffusion is recovered, with a surprising non analytical dependence of the
diffusion coefficient on the particle density. When the TP is allowed to visit
the teeth of the comb, based on a mean-field-like Continuous Time Random Walk
description, we unveil a rich and complex scenario, with several successive
subdiffusive regimes, resulting from the coupling between the inhomogeneous
comb geometry and particle interactions. Remarkably, the presence of hard-core
interactions speeds up the TP motion along the backbone of the structure in all
regimes.Comment: 5 pages, 3 figures + supplemental materia
Current-mediated synchronization of a pair of beating non-identical flagella
The basic phenomenology of experimentally observed synchronization (i.e., a
stochastic phase locking) of identical, beating flagella of a biflagellate alga
is known to be captured well by a minimal model describing the dynamics of
coupled, limit-cycle, noisy oscillators (known as the noisy Kuramoto model). As
demonstrated experimentally, the amplitudes of the noise terms therein, which
stem from fluctuations of the rotary motors, depend on the flagella length.
Here we address the conceptually important question which kind of synchrony
occurs if the two flagella have different lengths such that the noises acting
on each of them have different amplitudes. On the basis of a minimal model,
too, we show that a different kind of synchrony emerges, and here it is
mediated by a current carrying, steady-state; it manifests itself via
correlated "drifts" of phases. We quantify such a synchronization mechanism in
terms of appropriate order parameters and - for an ensemble of
trajectories and for a single realization of noises of duration ,
respectively. Via numerical simulations we show that both approaches become
identical for long observation times . This reveals an ergodic
behavior and implies that a single-realization order parameter is
suitable for experimental analysis for which ensemble averaging is not always
possible.Comment: 10 pages, 2 figure
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