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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 48 (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.
An example of BrunetDerrida behavior for a branchingselection particle system on Z. arXiv:0810.5567
, 2008
"... Abstract. We consider a class of branchingselection particle systems on R similar to the one considered by E. Brunet and B. Derrida in their 1997 paper ”Shift in the velocity of a front due to a cutoff”. Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the ..."
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Cited by 24 (0 self)
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Abstract. We consider a class of branchingselection particle systems on R similar to the one considered by E. Brunet and B. Derrida in their 1997 paper ”Shift in the velocity of a front due to a cutoff”. Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size N of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate (log N) −2. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of N independent branching random walks killed below a linear spacetime barrier. 1.
A new model for evolution in a spatial continuum
"... o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL ..."
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o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL
Effect of noise on front propagation in reactiondiffusion equations of KPP type. Preprint, available at arXiv:0902.3423
"... We consider reactiondiffusion equations of KPP type in one spatial dimension, perturbed by a FisherWright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed FisherKPP equations and ∂tu = ∂ 2 xu + u(1 − u) + ǫ p u(1 − u) ˙ W, (0.1) ∂tu = ∂ 2 ..."
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Cited by 15 (2 self)
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We consider reactiondiffusion equations of KPP type in one spatial dimension, perturbed by a FisherWright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed FisherKPP equations and ∂tu = ∂ 2 xu + u(1 − u) + ǫ p u(1 − u) ˙ W, (0.1) ∂tu = ∂ 2 xu + u(1 − u) + ǫ √ u ˙ W, (0.2) where ˙ W = ˙ W(t, x) is a spacetime white noise.
Effects of fluctuations on propagating fronts
 Physics ReportsReview Section of Physics Letters
, 2004
"... Propagating fronts are seen in varieties of nonequilibrium pattern forming systems in Physics, Chemistry and Biology. In the last two decades, many researchers have contributed to the understanding of the underlying dynamics of the propagating fronts. Of these, the deterministic and meanfield dyna ..."
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Propagating fronts are seen in varieties of nonequilibrium pattern forming systems in Physics, Chemistry and Biology. In the last two decades, many researchers have contributed to the understanding of the underlying dynamics of the propagating fronts. Of these, the deterministic and meanfield dynamics of the fronts were mostly understood in late 1980s and 1990s. On the other hand, although the earliest work on the effect of fluctuations on propagating fronts dates back to early 1980s, the subject of fluctuating fronts did not reach its adolescence until the mid 1990s. From there onwards the last few years witnessed a surge in activities in the effect of fluctuations on propagating fronts. Scores of papers have been written on this subject since then, contributing to a significant maturity of our understanding, and only recently a full picture of fluctuating fronts has started to emerge. This review is an attempt to collect all the works on fluctuating (propagating) fronts in a coherent and cogent manner in proper perspective. It is based on the idea of making our knowledge in this field available to a broader audience, and it is also expected to help to collect
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
A note on the rightmost particle in a FlemingViot process, ArXiv eprints
 EJP
, 2012
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BRUNETDERRIDA BEHAVIOR OF BRANCHINGSELECTION PARTICLE SYSTEMS ON THE LINE
, 811
"... Brunet and B. Derrida ”Shift in the velocity of a front due to a cutoff ” (see the reference within the paper), where it is shown, based on numerical simulations and heuristic arguments, that a certain branchingselection particle system on the line exhibits the following behavior: as N goes to infi ..."
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Brunet and B. Derrida ”Shift in the velocity of a front due to a cutoff ” (see the reference within the paper), where it is shown, based on numerical simulations and heuristic arguments, that a certain branchingselection particle system on the line exhibits the following behavior: as N goes to infinity, the asymptotic velocity of the system with N particles converges to a limiting value at the surprisingly slow rate (log N) −2. In this paper, we consider a class of branchingselection particle systems on R with N particles, defined through iterated branchingselection steps of the following type. 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 some distribution p. 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. Under generic assumptions on p, it is shown that BrunetDerrida behavior holds for the corresponding particle system. The proofs are based on ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and rely on a comparison of the particle system with a family of N independent branching random walks killed below a linear spacetime barrier. The results presented here both improve and generalize upon previous work by the first author of this paper, which was completed just before the results by Gantert, Hu and Shi became publicly available. 1.
Abstract Available online at www.sciencedirect.com Physics Reports 386 (2003) 29–222
, 2003
"... Front propagation into unstable states ..."
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