laboratoire de physique statistique
laboratoire de physique statistique


The number of accessible paths in the hypercube - Berestycki, Julien and Brunet, Eric and Shi, Zhan
BERNOULLI 22653-680 (2016) 

Abstract : Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube \0,1\(L) where each node carries an independent random variable uniformly distributed on [0, 1], except (1, 1,..., 1) which carries the value 1 and (0, 0,..., 0) which carries the value x is an element of [0, 1]. We study the number Theta of paths from vertex (0, 0,, 0) to the opposite vertex (1, 1,..., 1) along which the values on the nodes form an increasing sequence. We show that if the value on (0, 0,..., 0) is set to x = X/L then Theta/L converges in law as L -> infinity to e(-X) times the product of two standard independent exponential variables. As a first step in the analysis, we study the same question when the graph is that of a tree where the root has arity L, each node at level 1 has arity L - 1,..., and the nodes at level L - 1 have only one offspring which are the leaves of the tree (all the leaves are assigned the value 1, the root the value x is an element of [0, 11).
Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential - Berestycki, Julien and Brunet, Eric and Harris, John W. and Harris, Simon C. and Roberts, Matthew I.

Abstract : We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power p, for p is an element of [0, 2). The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following ``difficult'' paths, growth rates along ``easy'' paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate. Crown Copyright (C) 2015 Published by Elsevier B.V.
Genealogies in simple models of evolution - Brunet, Eric and Derrida, Bernard

Abstract : We review the statistical properties of the genealogies of a few models of evolution. In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population, in contrast with the linear growth of the neutral case. Moreover for a whole class of models, the statistics of the genealogies are those of the Bolthausen-Sznitman coalescent rather than the Kingman coalescent in the neutral case. For sexual reproduction in the neutral case, the time to reach the first common ancestors for the whole population and the time for all individuals to have all their ancestors in common are also logarithmic in the population size, as predicted by Chang in 1999. We discuss how these times are modified by introducing selection in a simple way.
Branching Brownian motion seen from its tip - Aidekon, E. and Berestycki, J. and Brunet, E. and Shi, Z.

Abstract : It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052-1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, A branching random walk seen from the tip, 2010, Poissonian statistics in the extremal process of branching Brownian motion, 2010; Arguin et al., The extremal process of branching Brownian motion, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (The extremal process of branching Brownian motion, 2011).
How genealogies are affected by the speed of evolution - Brunet, Eric and Derrida, Bernard

Abstract : In a series of recent works it has been shown that a class of simple models of evolving populations under selection leads to genealogical trees whose statistics are given by the Bolthausen-Sznitman coalescent rather than by the well-known Kingman coalescent in the case of neutral evolution. Here we show that when conditioning the genealogies on the speed of evolution, one finds a one-parameter family of tree statistics which interpolates between the Bolthausen-Sznitman and Kingman coalescents. This interpolation can be calculated explicitly for one specific version of the model, the exponential model. Numerical simulations of another version of the model and a phenomenological theory indicate that this one-parameter family of tree statistics could be universal. We compare this tree structure with those appearing in other contexts, in particular in the mean field theory of spin glasses.
A Branching Random Walk Seen from the Tip - Brunet, Eric and Derrida, Bernard

Abstract : We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. The distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.
Fluctuations of the heat flux of a one-dimensional hard particle gas - Brunet, E. and Derrida, B. and Gerschenfeld, A.
EPL 90 (2010) 

Abstract : Momentum-conserving one-dimensional models are known to exhibit anomalous Fourier's law, with a thermal conductivity varying as a power law of the system size. Here we measure, by numerical simulations, several cumulants of the heat flux of a one-dimensional hard particle gas. We find that the cumulants, like the conductivity, vary as power laws of the system size. Our results also indicate that cumulants higher than the second follow different power laws when one compares the ring geometry at equilibrium and the linear case in contact with two heat baths (at equal or unequal temperatures). Copyright (C) EPLA, 2010
The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential - Berestycki, J. and Brunet, E. and Harris, J. W. and Harris, S. C.

Abstract : In this note we consider a branching Brownian motion (BBM) on R in which a particle at spatial position y splits into two at rate beta y(2), where beta > 0 is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the population size remains finite, almost surely, for all time. We find an asymptotic for the almost-sure rate of growth of the population. (C) 2010 Elsevier B.V. All rights reserved.
Statistics at the tip of a branching random walk and the delay of traveling waves - Brunet, E. and Derrida, B.
EPL 87 (2009) 

Abstract : We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean-field spin-glasses. Copyright (C) EPLA, 2009
The traveling-wave approach to asexual evolution: Muller's ratchet and speed of adaptation - Rouzine, Igor M. and Brunet, Eric and Wilke, Claus O.

Abstract : We use traveling-wave theory to derive expressions for the rate of accumulation of deleterious mutations under Muller's ratchet and the speed of adaptation under positive selection in asexual populations. Traveling-wave theory is a semi-deterministic description of an evolving population, where the bulk of the population is modeled using deterministic equations, but the class of the highest-fitness genotypes, whose evolution over time determines loss or gain of fitness in the population, is given proper stochastic treatment. We derive improved methods to model the highest-fitness class (the stochastic edge) for both Muller's ratchet and adaptive evolution, and calculate analytic correction terms that compensate for inaccuracies which arise when treating discrete fitness classes as a continuum. We show that traveling-wave theory makes excellent predictions for the rate of mutation accumulation in the case of Muller's ratchet, and makes good predictions for the speed of adaptation in a very broad parameter range. We predict the adaptation rate to grow logarithmically in the population size until the population size is extremely large. (C) 2007 Elsevier Inc. All rights reserved.
The stochastic edge in adaptive evolution - Brunet, Eric and Rouzine, Igor M. and Wilke, Claus O.
GENETICS 179603-620 (2008) 

Abstract : In a recent article, Desai and Fisher proposed that the speed of adaptation in an asexual population is determined by the dynamics of the stochastic edge of the population, that is, by the emergence and subsequent establishment of rare mutants that exceed the fitness of all sequences currently present in the population. Desai and Fisher perform an elaborate stochastic calculation of the mean time tau until a new class of mutants has been established and interpret 1/tau as the speed of adaptation. As they note, however, their calculations are valid only for moderate speeds. This limitation arises from their method to determine T: Desai and Fisher back extrapolate the value of T from the best-fit class's exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows nonexponentially during the relevant time interval. Here, we substantially extend Desai and Fisher's analysis of the stochastic edge. We show that we can apply Desai and Fisher's method to high speeds by either exponentially back extrapolating from finite time or using a nonexponential back extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al.) and agree well with numerical simulations.
Universal tree structures in directed polymers and models of evolving populations - Brunet, Eric and Derrida, Bernard and Simon, Damien

Abstract : By measuring or calculating coalescence times for several models of coalescence or evolution, with and without selection, we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes.
Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization - Brunet, E. and Derrida, B. and Mueller, A. H. and Munier, S.

Abstract : We consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves. For one particular model that we shall call the exponential model, the properties of the traveling wave front can be calculated exactly, as well as the statistics of the genealogy of the population. One striking result is that, for this particular model, the genealogical trees have the same statistics as the trees of replicas in the Parisi mean-field theory of spin glasses. We also find that in the exponential model, the coalescence times along these trees grow like the logarithm of the population size. A phenomenological picture of the propagation of wave fronts that we introduced in a previous work, as well as our numerical data, suggest that these statistics remain valid for a larger class of models, while the coalescence times grow like the cube of the logarithm of the population size.