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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 ..."
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Cited by 45 (3 self)
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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.
BrunetDerrida particle systems, free boundary problems and WienerHopf equations
, 2009
"... We consider a branchingselection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in di ..."
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Cited by 11 (1 self)
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We consider a branchingselection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measurevalued process whose densities solve a free boundary integrodifferential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c ≥ a or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of WienerHopf equations. 1 Introduction and statement of the results We will consider the following branchingselection particle system. At any time t we have N particles on the real line with positions η N t (1) ≥ · · · ≥ η N t (N). Each one of the N particles gives birth at rate 1 to a new particle whose position is chosen, relative to the
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 ..."
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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
Evolution in predatorprey systems
, 2010
"... We study the adaptive dynamics of predator prey systems modeled by a dynamical system in which the characteristics are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexiste ..."
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Cited by 5 (2 self)
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We study the adaptive dynamics of predator prey systems modeled by a dynamical system in which the characteristics are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the parameters of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied we see (i) the number of coexisting predators remains tight and the differences of the parameters from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we conjecture that the differences converge to a deterministic limit. Key words: predatorprey, adaptive dynamics, branching random walk, branchingselection particle system 2000 MSC: primary 92D15, 92D25; secondary 60J60, 60K35 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 ..."
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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.
Front velocity and directed polymers in random medium
 Stochastic Processes and their Applications
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LAST PASSAGE PERCOLATION AND TRAVELING WAVES
"... Abstract. We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida [5]. The particles can be interpreted as last passage times in directed percolation on {1,..., N} of meanfield type. The particles remain grouped and move like a traveling wave, subject to disc ..."
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Abstract. We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida [5]. The particles can be interpreted as last passage times in directed percolation on {1,..., N} of meanfield type. The particles remain grouped and move like a traveling wave, subject to discretization and driven by a random noise. As N increases, we obtain estimates for the speed of the front and its profile, for different laws of the driving noise. The Gumbel distribution plays a central role for the particle jumps, and we show that the scaling limit is a Lévy process in this case. The case of bounded jumps yields a completely different behavior. 1. Definition of the model We consider the following stochastic process introduced by Brunet and Derrida [5]. It consists in a constant number N ≥ 1 of particles on the real line, initially at the positions X1(0),..., XN(0). Then, given the positions X1(t),..., XN(t) of the particles at time t ∈ N, we define the positions at time t + 1 by Xi(t + 1) = max Xj(t) + ξi,j(t + 1)
unknown title
, 2011
"... Rigorous results for the minimal speed of Kolmogorov–Petrovskii–Piscounov monotonic fronts with a cutoff ..."
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Rigorous results for the minimal speed of Kolmogorov–Petrovskii–Piscounov monotonic fronts with a cutoff