Results 1  10
of
12
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
Abstract

Cited by 46 (3 self)
 Add to MetaCart
Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
AN EXAMPLE OF BRUNETDERRIDA BEHAVIOR FOR A BRANCHINGSELECTION PARTICLE SYSTEM ON Z
, 2008
"... We consider a branchingselection particle system on Z with N ≥ 1 particles. During a branching step, each particle is replaced by two new particles, whose positions are shifted from that of the original particle by independently performing two random walk steps according to the distribution pδ1 + ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
We consider a branchingselection particle system on Z with N ≥ 1 particles. During a branching step, each particle is replaced by two new particles, whose positions are shifted from that of the original particle by independently performing two random walk steps according to the distribution pδ1 + (1 − p)δ0, from the location of the original particle. During the selection step that follows, only the N rightmost particles are kept among the 2N particles obtained at the branching step, to form a new population of N particles. After a large number of iterated branchingselection steps, the displacement of the whole population of N particles is ballistic, with deterministic asymptotic speed vN(p). As N goes to infinity, vN(p) converges to a finite limit v∞(p). Our main result is that, for every 0 < p < 1/2, as N goes to infinity, the order of magnitude of the difference v∞(p) − vN(p) is log(N) −2. This is called BrunetDerrida behavior in reference to the 1997 paper by E. Brunet and B. Derrida ”Shift in the velocity of a front due to a cutoff ” (see the reference within the paper), where such a behavior is established for a similar branchingselection particle system, using both numerical simulations and heuristic arguments.
The Critical Barrier for the Survival of the Branching Random Walk with Absorption
, 2009
"... We study a branching random walk on R with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. [4] determined whether a linear barrier allows the p ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
(Show Context)
We study a branching random walk on R with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. [4] determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term an 1/3 to the position of the barrier for the n th generation and find an explicit critical value ac such that the process dies when a < ac and survives when a> ac. We also obtain the rate of extinction when a < ac and a lower bound for the population when it survives. 1
RANDOM WALKS & TREES
"... These notes provide an elementary and selfcontained introduction to branching random walks. Chapter 1 gives a brief overview of Galton–Watson trees, whereas Chapter 2 presents the classical law of large numbers for branching random walks. These two short chapters are not exactly indispensable, but ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
These notes provide an elementary and selfcontained introduction to branching random walks. Chapter 1 gives a brief overview of Galton–Watson trees, whereas Chapter 2 presents the classical law of large numbers for branching random walks. These two short chapters are not exactly indispensable, but they introduce the idea of using sizebiased trees, thus giving motivations and an avantgoût to the main part, Chapter 3, where branching random walks are studied from a deeper point of view, and are connected to the model of directed polymers on a tree. Treerelated random processes form a rich and exciting research subject. These notes cover only special topics. For a general account, we refer to the StFlour lecture notes of Peres [47] and to the forthcoming book of Lyons and Peres [42], as well as to Duquesne and Le Gall [23] and Le Gall [37] for continuous random trees. I am grateful to the organizers of the Symposium for the kind invitation, and to my coauthors for sharing the pleasure of random climbs. Contents 1 Galton–Watson trees 1
Total Progeny in Killed Branching Random Walk
, 2009
"... We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of u ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of unkilled particles is almost surely finite [26, 31]. In this paper, we confirm a conjecture of Aldous [3, 4] that E [Z] < ∞ while E [Z log Z] = ∞. The proofs rely on precise large deviations estimates and ballot theoremstyle results for the sample paths of random walks. 1
Survival of homogenous fragmentation processes with killing
, 2011
"... We consider a homogenous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the growth of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
We consider a homogenous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the growth of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.
Critical branching Brownian motion with absorption: survival probability. Probab. Theory Relat. Fields
, 2013
"... Abstract We consider critical branching Brownian motion with absorption, in which there is initially a single particle at x > 0, particles move according to independent onedimensional Brownian motions with the critical drift of − √ 2, and particles are absorbed when they reach zero. Here we obt ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract We consider critical branching Brownian motion with absorption, in which there is initially a single particle at x > 0, particles move according to independent onedimensional Brownian motions with the critical drift of − √ 2, 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 x 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 rightmost particle. We also obtain asymptotic results for the configuration of particles at a typical time.