Dynamical aspects of a moving front model of mean-field type

Abstract : We focus on the discrete-time stochastic model studied by E. Brunet and B. Derrida in 2004: a fixed number $N$ of particles evolve on the real line according to a branching/selection mechanism. The particles remain grouped and move like a travelling-front driven by a random noise. Besides its the mathematical interest, moving fronts describe, for example, the evolution of systems having two different species $X$ and $Y$ of particles, reacting according to the irreversible auto-catalytic rule $X+Y \to 2X$. The model here is of mean-field type and the particles can also be interpreted as the last passage time in directed percolation on $\{1, \ldots, N\}$. It has been proved by F. Comets, J. Quastel and A. Ram\'irez in 2013 that the front moves globally at a deterministic \emph{speed} and that fluctuation occur in the diffusive scale $\sqrt{t}$. In this thesis, we compute the asymptotic speed as $N \to \infty$ for a large class of random disorders. We prove that the finite-size correction to the \emph{speed} satisfies \emph{universal features} depending on the upper-tail probabilities. For a certain class of noise, the techniques we have developed also allow to compute the asymptotic diffusion constant. From a different perspective, one can also interpret the model as the dynamics of a constant size population, the positions being the fitnesses of the individuals. In this case, we focus on how individuals are related and how many generation one has to go back in time in order to find a common ancestor. For a specific choice of disorder, we show that the average coalescence times scale like $\ln N$ and that the limit genealogical trees are governed by the Bolthausen-Sznitman coalescent, which validates the physics predictions for this class of models.
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Submitted on : Wednesday, October 7, 2015 - 10:48:56 PM
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Aser Cortines. Dynamical aspects of a moving front model of mean-field type. Mathematics [math]. Paris 7, 2015. English. ⟨tel-01213197⟩

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