515 research outputs found
Survival probability of the branching random walk killed below a linear boundary
We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on
the asymptotic behavior of the survival probability of the branching random
walk killed below a linear boundary, in the special case of deterministic
binary branching and bounded random walk steps. Connections with the
Brunet-Derrida theory of stochastic fronts are discussed
Coupling times with ambiguities for particle systems and applications to context-dependent DNA substitution models
We define a notion of coupling time with ambiguities for interacting particle
systems, and show how this can be used to prove ergodicity and to bound the
convergence time to equilibrium and the decay of correlations at equilibrium. A
motivation is to provide simple conditions which ensure that perturbed particle
systems share some properties of the underlying unperturbed system. We apply
these results to context-dependent substitution models recently introduced by
molecular biologists as descriptions of DNA evolution processes. These models
take into account the influence of the neighboring bases on the substitution
probabilities at a site of the DNA sequence, as opposed to most usual
substitution models which assume that sites evolve independently of each other.Comment: 33 page
Coupling from the past times with ambiguities and perturbations of interacting particle systems
We discuss coupling from the past techniques (CFTP) for perturbations of
interacting particle systems on the d-dimensional integer lattice, with a
finite set of states, within the framework of the graphical construction of the
dynamics based on Poisson processes. We first develop general results for what
we call CFTP times with ambiguities. These are analogous to classical coupling
(from the past) times, except that the coupling property holds only provided
that some ambiguities concerning the stochastic evolution of the system are
resolved. If these ambiguities are rare enough on average, CFTP times with
ambiguities can be used to build actual CFTP times, whose properties can be
controlled in terms of those of the original CFTP time with ambiguities. We
then prove a general perturbation result, which can be stated informally as
follows. Start with an interacting particle system possessing a CFTP time whose
definition involves the exploration of an exponentially integrable number of
points in the graphical construction, and which satisfies the positive rates
property. Then consider a perturbation obtained by adding new transitions to
the original dynamics. Our result states that, provided that the perturbation
is small enough (in the sense of small enough rates), the perturbed interacting
particle system too possesses a CFTP time (with nice properties such as an
exponentially decaying tail). The proof consists in defining a CFTP time with
ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed
dynamics. Finally, we discuss examples of particle systems to which this result
can be applied. Concrete examples include a class of neighbor-dependent
nucleotide substitution model, and variations of the classical voter model,
illustrating the ability of our approach to go beyond the case of weakly
interacting particle systems.Comment: This paper is an extended and revised version of an earlier
manuscript available as arXiv:0712.0072, where the results were limited to
perturbations of RN+YpR nucleotide substitution model
Fluctuations of the front in a one-dimensional model for the spread of an infection
We study the following microscopic model of infection or epidemic reaction:
red and blue particles perform independent nearest-neighbor continuous-time
symmetric random walks on the integer lattice with jump rates
for red particles and for blue particles, the interaction rule
being that blue particles turn red upon contact with a red particle. The
initial condition consists of i.i.d. Poisson particle numbers at each site,
with particles at the left of the origin being red, while particles at the
right of the origin are blue. We are interested in the dynamics of the front,
defined as the rightmost position of a red particle. For the case ,
Kesten and Sidoravicius established that the front moves ballistically, and
more precisely that it satisfies a law of large numbers. Their proof is based
on a multi-scale renormalization technique, combined with approximate
sub-additivity arguments. In this paper, we build a renewal structure for the
front propagation process, and as a corollary we obtain a central limit theorem
for the front when . Moreover, this result can be extended to the case
where , up to modifying the dynamics so that blue particles turn red
upon contact with a site that has previously been occupied by a red particle.
Our approach extends the renewal structure approach developed by Comets,
Quastel and Ram\'{{\i}}rez for the so-called frog model, which corresponds to
the case.Comment: Published at http://dx.doi.org/10.1214/15-AOP1034 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Solvable models of neighbor-dependent nucleotide substitution processes
We prove that a wide class of models of Markov neighbor-dependent
substitution processes on the integer line is solvable. This class contains
some models of nucleotide substitutions recently introduced and studied
empirically by molecular biologists. We show that the polynucleotide
frequencies at equilibrium solve explicit finite-size linear systems. Finally,
the dynamics of the process and the distribution at equilibrium exhibit some
stringent, rather unexpected, independence properties. For example, nucleotide
sites at distance at least three evolve independently, and the sites, if
encoded as purines and pyrimidines, evolve independently.Comment: 47 pages, minor modification
An example of Brunet-Derrida behavior for a branching-selection particle system on
We consider a branching-selection particle system on with
particles. During a branching step, each particle is replaced by two new
particles, whose positions are shifted from that of the original particle by
independently performing two random walk steps according to the distribution , from the location of the original particle.
During the selection step that follows, only the N rightmost particles are kept
among the 2N particles obtained at the branching step, to form a new population
of particles. After a large number of iterated branching-selection steps,
the displacement of the whole population of particles is ballistic, with
deterministic asymptotic speed . As goes to infinity,
converges to a finite limit . The main result is that, for every
, as goes to infinity, the order of magnitude of the difference
is . This is called Brunet-Derrida
behavior in reference to the 1997 paper by E. Brunet and B. Derrida "Shift in
the velocity of a front due to a cutoff" (see the reference within the paper),
where such a behavior is established for a similar branching-selection particle
system, using both numerical simulations and heuristic arguments
Brunet-Derrida behavior of branching-selection particle systems on the line
We consider a class of branching-selection particle systems on similar
to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in
the velocity of a front due to a cutoff". Based on numerical simulations and
heuristic arguments, Brunet and Derrida showed that, as the population size
of the particle system goes to infinity, the asymptotic velocity of the system
converges to a limiting value at the unexpectedly slow rate . In
this paper, we give a rigorous mathematical proof of this fact, for the class
of particle systems we consider. The proof makes use of ideas and results by R.
Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of
the particle system with a family of independent branching random walks
killed below a linear space-time barrier
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