56 research outputs found
The hyperbolic geometry of random transpositions
Turn the set of permutations of objects into a graph by connecting
two permutations that differ by one transposition, and let be the
simple random walk on this graph. In a previous paper, Berestycki and Durrett
[In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of
the distance from the identity at time has a phase transition at .
Here we investigate some consequences of this result for the geometry of .
Our first result can be interpreted as a breakdown for the Gromov hyperbolicity
of the graph as seen by the random walk, which occurs at a critical radius
equal to . Let be a triangle formed by the origin and two points
sampled independently from the hitting distribution on the sphere of radius
for a constant . Then when , if the geodesics are suitably
chosen, with high probability is -thin for some , whereas
it is always O(n)-thick when . We also show that the hitting
distribution of the sphere of radius is asymptotically singular with
respect to the uniform distribution. Finally, we prove that the critical
behavior of this Gromov-like hyperbolicity constant persists if the two
endpoints are sampled from the uniform measure on the sphere of radius .
However, in this case, the critical radius is .Comment: Published at http://dx.doi.org/10.1214/009117906000000043 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Effect of scale on long-range random graphs and chromosomal inversions
We consider bond percolation on vertices on a circle where edges are
permitted between vertices whose spacing is at most some number L=L(n). We show
that the resulting random graph gets a giant component when
(when the mean degree exceeds 1) but not when . The proof uses
comparisons to branching random walks. We also consider a related process of
random transpositions of particles on a circle, where transpositions only
occur again if the spacing is at most . Then the process exhibits the
mean-field behavior described by Berestycki and Durrett if and only if L(n)
tends to infinity, no matter how slowly. Thus there are regimes where the
random graph has no giant component but the random walk nevertheless has a
phase transition. We discuss possible relevance of these results for a dataset
coming from D. repleta and D. melanogaster and for the typical length of
chromosomal inversions.Comment: Published in at http://dx.doi.org/10.1214/11-AAP793 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mixing times for random k-cycles and coalescence-fragmentation chains
Let be the permutation group on elements, and consider a
random walk on whose step distribution is uniform on
-cycles. We prove a well-known conjecture that the mixing time of this
process is , with threshold of width linear in . Our proofs
are elementary and purely probabilistic, and do not appeal to the
representation theory of .Comment: Published in at http://dx.doi.org/10.1214/10-AOP634 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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