1,196 research outputs found
Weak law of large numbers for some Markov chains along non homogeneous genealogies
We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A Markov chain models the dynamic of the trait of each individual along this genealogy and may also be time non-homogeneous. Such models are motivated by transmission processes in the cell division, reproduction-dispersion dynamics or sampling problems in evolution. We want to determine the evolution of the distribution of the traits among the population, namely the asymptotic behavior of the proportion of individuals with a given trait. We prove some quenched laws of large numbers which rely on the ergodicity of an auxiliary process, in the same vein as \cite{guy,delmar}. Applications to time inhomogeneous Markov chains lead us to derive a backward (with respect to the environment) law of large numbers and a law of large numbers on the whole population until generation . A central limit is also established in the transient case
Strict monotonicity properties in one-dimensional excited random walks
We consider one-dimensional excited random walks with finitely many cookies
at each site. There are certain natural monotonicity results that are known for
the excited random walk under some partial orderings of the cookie
environments. We improve these monotonicity results to be strictly monotone
under a partial ordering of cookie environments introduced by Holmes and
Salisbury. While the self-interacting nature of the excited random walk makes a
direct coupling proof difficult, we show that there is a very natural coupling
of the associated branching process from which the monotonicity results follow
Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process
Non-trivial linear bounds are obtained for the displacement of a random walk
in a dynamic random environment given by a one-dimensional simple symmetric
exclusion process in equilibrium. The proof uses an adaptation of multiscale
renormalization methods of Kesten and Sidoravicius.Comment: 20 pages, 3 figure
Ecological equilibrium for restrained branching random walks
We study a generalized branching random walk where particles breed at a rate
which depends on the number of neighboring particles. Under general assumptions
on the breeding rates we prove the existence of a phase where the population
survives without exploding. We construct a nontrivial invariant measure for
this case.Comment: Published in at http://dx.doi.org/10.1214/105051607000000203 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Diffusion-limited annihilation in inhomogeneous environments
We study diffusion-limited (on-site) pair annihilation and
(on-site) fusion which we show to be equivalent for arbitrary
space-dependent diffusion and reaction rates. For one-dimensional lattices with
nearest neighbour hopping we find that in the limit of infinite reaction rate
the time-dependent -point density correlations for many-particle initial
states are determined by the correlation functions of a dual diffusion-limited
annihilation process with at most particles initially. By reformulating
general properties of annihilating random walks in one dimension in terms of
fermionic anticommutation relations we derive an exact representation for these
correlation functions in terms of conditional probabilities for a single
particle performing a random walk with dual hopping rates. This allows for the
exact and explicit calculation of a wide range of universal and non-universal
types of behaviour for the decay of the density and density correlations.Comment: 27 pages, Latex, to appear in Z. Phys.
- …