32 research outputs found
Ruelle's probability cascades seen as a fragmentation process
In this paper, we study Ruelle's probability cascades in the framework of
time-inhomogeneous fragmentation processes. We describe Ruelle's cascades
mechanism exhibiting a family of measures that
characterizes its infinitesimal evolution. To this end, we will first extend
the time-homogeneous fragmentation theory to the inhomogeneous case. In the
last section, we will study the behavior for small and large times of Ruelle's
fragmentation process
On the equivalence of some eternal additive coalescents
In this paper, we study additive coalescents. Using their representation as
fragmentation processes, we prove that the law of a large class of eternal
additive coalescents is absolutely continuous with respect to the law of the
standard additive coalescent on any bounded time interval
Fragmentation of compositions and intervals
The fragmentation processes of exchangeable partitions have already been
studied by several authors. In this paper, we examine rather fragmentation of
exchangeable compositions, that means partitions of where the
order of the blocks counts. We will prove that such a fragmentation is
bijectively associated to an interval fragmentation. Using this correspondence,
we then calculate the Hausdorff dimension of certain random closed set that
arise in interval fragmentations and we study Ruelle's interval fragmentation
Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
We work in the context of the infinitely many alleles model. The allelic
partition associated with a coalescent process started from n individuals is
obtained by placing mutations along the skeleton of the coalescent tree; for
each individual, we trace back to the most recent mutation affecting it and
group together individuals whose most recent mutations are the same. The number
of blocks of each of the different possible sizes in this partition is the
allele frequency spectrum. The celebrated Ewens sampling formula gives precise
probabilities for the allele frequency spectrum associated with Kingman's
coalescent. This (and the degenerate star-shaped coalescent) are the only
Lambda coalescents for which explicit probabilities are known, although they
are known to satisfy a recursion due to Moehle. Recently, Berestycki,
Berestycki and Schweinsberg have proved asymptotic results for the allele
frequency spectra of the Beta(2-alpha,alpha) coalescents with alpha in (1,2).
In this paper, we prove full asymptotics for the case of the
Bolthausen-Sznitman coalescent.Comment: 26 pages, 1 figur
Continuous-time vertex reinforced jump processes on Galton-Watson trees
We consider a continuous-time vertex reinforced jump process on a
supercritical Galton-Watson tree. This process takes values in the set of
vertices of the tree and jumps to a neighboring vertex with rate proportional
to the local time at that vertex plus a constant . The walk is either
transient or recurrent depending on this parameter . In this paper, we
complete results previously obtained by Davis and Volkov [Probab. Theory
Related Fields 123 (2002) 281-300, Probab. Theory Related Fields 128 (2004)
42-62] and Collevecchio [Ann. Probab. 34 (2006) 870-878, Electron. J. Probab.
14 (2009) 1936-1962] by proving that there is a unique (explicit) positive
such that the walk is recurrent for and transient for .Comment: Published in at http://dx.doi.org/10.1214/11-AAP811 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the speed of a cookie random walk
We consider the model of the one-dimensional cookie random walk when the
initial cookie distribution is spatially uniform and the number of cookies per
site is finite. We give a criterion to decide whether the limiting speed of the
walk is non-zero. In particular, we show that a positive speed may be obtained
for just 3 cookies per site. We also prove a result on the continuity of the
speed with respect to the initial cookie distribution
Rate of growth of a transient cookie random walk
We consider a one-dimensional transient cookie random walk. It is known from
a previous paper that a cookie random walk has positive or zero speed
according to some positive parameter or . In this article,
we give the exact rate of growth of in the zero speed regime, namely:
for , converges in law to a
Mittag-Leffler distribution whereas for , converges
in probability to some positive constant
Distances in the highly supercritical percolation cluster
On the supercritical percolation cluster with parameter p, the distances
between two distant points of the axis are asymptotically increased by a factor
1+(1-p)/2+o(1-p) with respect to the usual distance. The proof is based on an
apparently new connection with the TASEP (totally asymmetric simple exclusion
process).Comment: 15 page