81 research outputs found
Exchangeable Fragmentation-Coalescence processes and their equilibrium measures
We define and study a family of Markov processes with state space the compact
set of all partitions of N that we call exchangeable fragmentation-coalescence
processes. They can be viewed as a combination of exchangeable fragmentation as
defined by Bertoin and of homogenous coalescence as defined by Pitman and
Schweinsberg or Mohle and Sagitov. We show that they admit a unique invariant
probability measure and we study some properties of their paths and of their
equilibrium measure
Critical branching Brownian motion with absorption: survival probability
We consider branching Brownian motion on the real line with absorption at
zero, in which particles move according to independent Brownian motions with
the critical drift of . Kesten (1978) showed that almost surely this
process eventually dies out. Here we obtain upper and lower bounds on the
probability that the process survives until some large time . These bounds
improve upon results of Kesten (1978), and partially confirm nonrigorous
predictions of Derrida and Simon (2007)
A small-time coupling between -coalescents and branching processes
We describe a new general connection between -coalescents and
genealogies of continuous-state branching processes. This connection is based
on the construction of an explicit coupling using a particle representation
inspired by the lookdown process of Donnelly and Kurtz. This coupling has the
property that the coalescent comes down from infinity if and only if the
branching process becomes extinct, thereby answering a question of Bertoin and
Le Gall. The coupling also offers new perspective on the speed of coming down
from infinity and allows us to relate power-law behavior for
to the classical upper and lower indices arising in the study of pathwise
properties of L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/12-AAP911 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The genealogy of branching Brownian motion with absorption
We consider a system of particles which perform branching Brownian motion
with negative drift and are killed upon reaching zero, in the near-critical
regime where the total population stays roughly constant with approximately N
particles. We show that the characteristic time scale for the evolution of this
population is of order , in the sense that when time is measured in
these units, the scaled number of particles converges to a variant of Neveu's
continuous-state branching process. Furthermore, the genealogy of the particles
is then governed by a coalescent process known as the Bolthausen-Sznitman
coalescent. This validates the nonrigorous predictions by Brunet, Derrida,
Muller and Munier for a closely related model.Comment: Published in at http://dx.doi.org/10.1214/11-AOP728 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Beta-coalescents and continuous stable random trees
Coalescents with multiple collisions, also known as -coalescents,
were introduced by Pitman and Sagitov in 1999. These processes describe the
evolution of particles that undergo stochastic coagulation in such a way that
several blocks can merge at the same time to form a single block. In the case
that the measure is the
distribution, they are also known to describe the genealogies of large
populations where a single individual can produce a large number of offspring.
Here, we use a recent result of Birkner et al. to prove that Beta-coalescents
can be embedded in continuous stable random trees, about which much is known
due to the recent progress of Duquesne and Le Gall. Our proof is based on a
construction of the Donnelly--Kurtz lookdown process using continuous random
trees, which is of independent interest. This produces a number of results
concerning the small-time behavior of Beta-coalescents. Most notably, we
recover an almost sure limit theorem of the present authors for the number of
blocks at small times and give the multifractal spectrum corresponding to the
emergence of blocks with atypical size. Also, we are able to find exact
asymptotics for sampling formulae corresponding to the site frequency spectrum
and the allele frequency spectrum associated with mutations in the context of
population genetics.Comment: Published in at http://dx.doi.org/10.1214/009117906000001114 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Small-time behavior of beta coalescents
For a finite measure on , the -coalescent is
a coalescent process such that, whenever there are clusters, each -tuple
of clusters merges into one at rate
. It has recently been shown
that if , the -coalescent in which is the
distribution can be used to describe
the genealogy of a continuous-state branching process (CSBP) with an
-stable branching mechanism. Here we use facts about CSBPs to establish
new results about the small-time asymptotics of beta coalescents. We prove an
a.s. limit theorem for the number of blocks at small times, and we establish
results about the sizes of the blocks. We also calculate the Hausdorff and
packing dimensions of a metric space associated with the beta coalescents, and
we find the sum of the lengths of the branches in the coalescent tree, both of
which are determined by the behavior of coalescents at small times. We extend
most of these results to other -coalescents for which
has the same asymptotic behavior near zero as the distribution. This work complements recent work of
Bertoin and Le Gall, who also used CSBPs to study small-time properties of
-coalescents.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP103 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Critical branching Brownian motion with absorption: particle configurations
We consider critical branching Brownian motion with absorption, in which
there is initially a single particle at , particles move according to
independent one-dimensional Brownian motions with the critical drift of
, and particles are absorbed when they reach zero. Here we obtain
asymptotic results concerning the behavior of the process before the extinction
time, as the position of the initial particle tends to infinity. We
estimate the number of particles in the system at a given time and the position
of the right-most particle. We also obtain asymptotic results for the
configuration of particles at a typical time
Large deviations for Branching Processes in Random Environment
A branching process in random environment is a
generalization of Galton Watson processes where at each generation the
reproduction law is picked randomly. In this paper we give several results
which belong to the class of {\it large deviations}. By contrast to the
Galton-Watson case, here random environments and the branching process can
conspire to achieve atypical events such as when is
smaller than the typical geometric growth rate and
when . One way to obtain such an atypical rate of growth is to have
a typical realization of the branching process in an atypical sequence of
environments. This gives us a general lower bound for the rate of decrease of
their probability. When each individual leaves at least one offspring in the
next generation almost surely, we compute the exact rate function of these
events and we show that conditionally on the large deviation event, the
trajectory converges to a
deterministic function in probability in the sense of
the uniform norm. The most interesting case is when and we
authorize individuals to have only one offspring in the next generation. In
this situation, conditionally on , the population size stays
fixed at 1 until a time . After time an atypical sequence
of environments let grow with the appropriate rate () to
reach The corresponding map is piecewise linear and is 0 on
and on $[t_c,1].
Accessibility percolation with backsteps
Consider a graph in which each site is endowed with a value called
\emph{fitness}. A path in the graph is said to be "open" or "accessible" if the
fitness values along that path is strictly increasing. We say that there is
accessibility percolation between two sites when such a path between them
exists. Motivated by the so called House-of-Cards model from evolutionary
biology, we consider this question on the -hypercube where the
fitness values are independent random variables. We show that, in the large
limit, the probability that an accessible path exists from an arbitrary
starting point to the (random) fittest site is no more than and we conjecture that this probability
does converge to . More precisely, there is a phase transition on
the value of the fitness of the starting site: assuming that the fitnesses
are uniform in , we show that, in the large limit, there is almost
surely no path to the fittest site if and we conjecture that
there are almost surely many paths if .
If one conditions on the fittest site to be on the opposite corner of the
starting site rather than being randomly chosen, the picture remains the same
but with the critical point being now .
Along the way, we obtain a large estimation for the number of self-avoiding
paths joining two opposite corners of the -hypercube
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