1,676 research outputs found
Exit times in non-Markovian drifting continuous-time random walk processes
By appealing to renewal theory we determine the equations that the mean exit
time of a continuous-time random walk with drift satisfies both when the
present coincides with a jump instant or when it does not. Particular attention
is paid to the corrections ensuing from the non-Markovian nature of the
process. We show that when drift and jumps have the same sign the relevant
integral equations can be solved in closed form. The case when holding times
have the classical Erlang distribution is considered in detail.Comment: 9 pages, 3 color plots, two-column revtex 4; new Appendix and
references adde
Stochastic Chemical Reactions in Micro-domains
Traditional chemical kinetics may be inappropriate to describe chemical
reactions in micro-domains involving only a small number of substrate and
reactant molecules. Starting with the stochastic dynamics of the molecules, we
derive a master-diffusion equation for the joint probability density of a
mobile reactant and the number of bound substrate in a confined domain. We use
the equation to calculate the fluctuations in the number of bound substrate
molecules as a function of initial reactant distribution. A second model is
presented based on a Markov description of the binding and unbinding and on the
mean first passage time of a molecule to a small portion of the boundary. These
models can be used for the description of noise due to gating of ionic channels
by random binding and unbinding of ligands in biological sensor cells, such as
olfactory cilia, photo-receptors, hair cells in the cochlea.Comment: 33 pages, Journal Chemical Physic
Random trees between two walls: Exact partition function
We derive the exact partition function for a discrete model of random trees
embedded in a one-dimensional space. These trees have vertices labeled by
integers representing their position in the target space, with the SOS
constraint that adjacent vertices have labels differing by +1 or -1. A
non-trivial partition function is obtained whenever the target space is bounded
by walls. We concentrate on the two cases where the target space is (i) the
half-line bounded by a wall at the origin or (ii) a segment bounded by two
walls at a finite distance. The general solution has a soliton-like structure
involving elliptic functions. We derive the corresponding continuum scaling
limit which takes the remarkable form of the Weierstrass p-function with
constrained periods. These results are used to analyze the probability for an
evolving population spreading in one dimension to attain the boundary of a
given domain with the geometry of the target (i) or (ii). They also translate,
via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main
modifications in Sect. 5-6 and conclusio
Analytical Solution of a Stochastic Content Based Network Model
We define and completely solve a content-based directed network whose nodes
consist of random words and an adjacency rule involving perfect or approximate
matches, for an alphabet with an arbitrary number of letters. The analytic
expression for the out-degree distribution shows a crossover from a leading
power law behavior to a log-periodic regime bounded by a different power law
decay. The leading exponents in the two regions have a weak dependence on the
mean word length, and an even weaker dependence on the alphabet size. The
in-degree distribution, on the other hand, is much narrower and does not show
scaling behavior. The results might be of interest for understanding the
emergence of genomic interaction networks, which rely, to a large extent, on
mechanisms based on sequence matching, and exhibit similar global features to
those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to
gene regulation networks, fixed minor errors and replaced fig. 4. Main body
of paper (model and calculations) remains unchanged. Submitted for
publicatio
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Minimal Absent Words in Prokaryotic and Eukaryotic Genomes
Minimal absent words have been computed in genomes of organisms from all domains of life. Here, we explore different sets of minimal absent words in the genomes of 22 organisms (one archaeota, thirteen bacteria and eight eukaryotes). We investigate if the mutational biases that may explain the deficit of the shortest absent words in vertebrates are also pervasive in other absent words, namely in minimal absent words, as well as to other organisms. We find that the compositional biases observed for the shortest absent words in vertebrates are not uniform throughout different sets of minimal absent words. We further investigate the hypothesis of the inheritance of minimal absent words through common ancestry from the similarity in dinucleotide relative abundances of different sets of minimal absent words, and find that this inheritance may be exclusive to vertebrates
Aging in the random energy model
In this letter we announce rigorous results on the phenomenon of aging in the
Glauber dynamics of the random energy model and their relation to Bouchaud's
'REM-like' trap model. We show that, below the critical temperature, if we
consider a time-scale that diverges with the system size in such a way that
equilibrium is almost, but not quite reached on that scale, a suitably defined
autocorrelation function has the same asymptotic behaviour than its analog in
the trap model.Comment: 4pp, P
First-passage and first-exit times of a Bessel-like stochastic process
We study a stochastic process related to the Bessel and the Rayleigh
processes, with various applications in physics, chemistry, biology, economics,
finance and other fields. The stochastic differential equation is , where is the Wiener process. Due to the
singularity of the drift term for , different natures of boundary at
the origin arise depending on the real parameter : entrance, exit, and
regular. For each of them we calculate analytically and numerically the
probability density functions of first-passage times or first-exit times.
Nontrivial behaviour is observed in the case of a regular boundary.Comment: 15 pages, 6 figures, submitted to Physical Review
Watermelon configurations with wall interaction: exact and asymptotic results
We perform an exact and asymptotic analysis of the model of vicious
walkers interacting with a wall via contact potentials, a model introduced by
Brak, Essam and Owczarek. More specifically, we study the partition function of
watermelon configurations which start on the wall, but may end at arbitrary
height, and their mean number of contacts with the wall. We improve and extend
the earlier (partially non-rigorous) results by Brak, Essam and Owczarek,
providing new exact results, and more precise and more general asymptotic
results, in particular full asymptotic expansions for the partition function
and the mean number of contacts. Furthermore, we relate this circle of problems
to earlier results in the combinatorial and statistical literature.Comment: AmS-TeX, 41 page
Hanbury Brown-Twiss interferometry and second-order correlations of inflaton quanta
The quantum theory of optical coherence is applied to the scrutiny of the
statistical properties of the relic inflaton quanta. After adapting the
description of the quantized scalar and tensor modes of the geometry to the
analysis of intensity correlations, the normalized degrees of first-order and
second-order coherence are computed in the concordance paradigm and are shown
to encode faithfully the statistical properties of the initial quantum state.
The strongly bunched curvature phonons are not only super-Poissonian but also
super-chaotic. Testable inequalities are derived in the limit of large angular
scales and can be physically interpreted in the light of the tenets of Hanbury
Brown-Twiss interferometry. The quantum mechanical results are compared and
contrasted with different situations including the one where intensity
correlations are the result of a classical stochastic process. The survival of
second-order correlations (not necessarily related to the purity of the initial
quantum state) is addressed by defining a generalized ensemble where
super-Poissonian statistics is an intrinsic property of the density matrix and
turns out to be associated with finite volume effects which are expected to
vanish in the thermodynamic limit.Comment: 42 pages, 3 included figures; corrected typos; to appear in Physical
Review
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