43 research outputs found
Subcritical regimes in the Poisson Boolean model of continuum percolation
We consider the Poisson Boolean model of continuum percolation. We show that
there is a subcritical phase if and only if is finite, where
denotes the radius of the balls around Poisson points and denotes the
dimension. We also give related results concerning the integrability of the
diameter of subcritical clusters.Comment: Published in at http://dx.doi.org/10.1214/07-AOP352 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Shape of territories in some competing growth models
We study two competing growth models. Each of these models describes the
spread of a finite number of infections on a graph. Each infection evolves like
an (oriented or unoriented) first passage percolation process except that once
a vertex is infected by type infection, it remains of type forever. We
give results about the shape of the area ultimately infected by the different
infections.Comment: Published in at http://dx.doi.org/10.1214/105051607000000113 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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
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