1,476 research outputs found
Percolation in an ultrametric space
We study percolation on the hierarchical lattice of order where the
probability of connection between two points separated by distance is of
the form . Since the distance is an
ultrametric, there are significant differences with percolation on the
Euclidean lattice. There are two non-critical regimes: , where
percolation occurs, and , where it does not occur. In the critical
case, , we use an approach in the spirit of the renormalization
group method of statistical physics and connectivity results of Erd\H{o}s-Renyi
random graphs play a key role. We find sufficient conditions on such that
percolation occurs, or that it does not occur. An intermediate situation called
pre-percolation is also considered. In the cases of percolation we prove
uniqueness of the constructed percolation clusters. In a previous paper
\cite{DG1} we studied percolation in the limit (mean field
percolation) which provided a simplification that allowed finding a necessary
and sufficient condition for percolation. For fixed there are open
questions, in particular regarding the existence of a critical value of a
parameter in the definition of , and if it exists, what would be the
behaviour at the critical point
Stochastic equations, flows and measure-valued processes
We first prove some general results on pathwise uniqueness, comparison
property and existence of nonnegative strong solutions of stochastic equations
driven by white noises and Poisson random measures. The results are then used
to prove the strong existence of two classes of stochastic flows associated
with coalescents with multiple collisions, that is, generalized Fleming--Viot
flows and flows of continuous-state branching processes with immigration. One
of them unifies the different treatments of three kinds of flows in Bertoin and
Le Gall [Ann. Inst. H. Poincar\'{e} Probab. Statist. 41 (2005) 307--333]. Two
scaling limit theorems for the generalized Fleming--Viot flows are proved,
which lead to sub-critical branching immigration superprocesses. From those
theorems we derive easily a generalization of the limit theorem for finite
point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006)
147--181].Comment: Published in at http://dx.doi.org/10.1214/10-AOP629 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotic behavior of the Poisson--Dirichlet distribution for large mutation rate
The large deviation principle is established for the Poisson--Dirichlet
distribution when the parameter approaches infinity. The result is
then used to study the asymptotic behavior of the homozygosity and the
Poisson--Dirichlet distribution with selection. A phase transition occurs
depending on the growth rate of the selection intensity. If the selection
intensity grows sublinearly in , then the large deviation rate function
is the same as the neutral model; if the selection intensity grows at a linear
or greater rate in , then the large deviation rate function includes an
additional term coming from selection. The application of these results to the
heterozygote advantage model provides an alternate proof of one of Gillespie's
conjectures in [Theoret. Popul. Biol. 55 145--156].Comment: Published at http://dx.doi.org/10.1214/105051605000000818 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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