Lectures notes (mostly in french)

• Thermodynamique et Physique Statistique en L3, module 3PY103
• Cours de Physique Statistique de M1, module 4P005
• Thermodynamique en L3, module 3P011. Thermodynamics (english translation)
• Initiation à la programmation en C
• Cours de remise à niveau en mathématiques élémentaires
• Research lecture on the stochastic Fisher-KPP front equation (given in Marseille 2007, in french)

Subjects

HDR

• The manuscript of my HDR, as a pdf file of 3.9 MB.
• The files used for my defense (slides and movies)

Publications

Papers in scientific journals

1. Éric Brunet and Bernard Derrida, Shift in the velocity of a front due to a cutoff, Physical Review E 1997, 56 (3), 2597-2604.
   arXiv, PRE, pdf.
   We consider the effect of a small cut-off ε on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off ε is to select a single velocity which converges when ε goes to 0 to the one predicted by the marginal stability argument. For small ε, the shift in velocity has the form K(log ε)^-2 and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.
   PACS numbers: 02.50.Ey, 03.40.Kf, 47.20.Ky
2. Éric Brunet and Bernard Derrida, Microscopic models of traveling wave equations, Computer Physics Communications, 1999 121-122, 376-381.
   arXiv, CPC, pdf.
   Reaction-diffusion problems are often described at a macroscopic scale by partial derivative equations of the type of the Fisher or Kolmogorov-Petrovsky-Piscounov equation. These equations have a continuous family of front solutions, each of them corresponding to a different velocity of the front. By simulating systems of size up to N=10¹⁶ particles at the microscopic scale, where particles react and diffuse according to some stochastic rules, we show that a single velocity is selected for the front. This velocity converges logarithmically to the solution of the F-KPP equation with minimal velocity when the number N of particles increases. A simple calculation of the effect introduced by the cutoff due to the microscopic scale allows one to understand the origin of the logarithmic correction.
   PACS numbers: 02.50.Ey, 03.40.Kf, 47.20.Ky
3. Éric Brunet and Bernard Derrida, Ground state energy of a non-integer number of particles with δ attractive interactions, Physica A 2000, 279 (1-4), 398-407.
   arXiv, Phys. A, pdf.
   We show how to define and calculate the ground state energy of a system of quantum particles with delta attractive interactions when the number of particles n is non-integer. The question is relevant to obtain the probability distribution of the free energy of a directed polymer in a random medium. When one expands the ground state energy in powers of the interaction, all the coefficients of the perturbation series are polynomials in n, allowing to define the perturbation theory for non-integer n. We develop a procedure to calculate all the cumulants of the free energy of the directed polymer and we give explicit, although complicated, expressions of the first three cumulants.
   PACS numbers: 05.40.+j, 02.50.-r, 82.20.-w.
4. Éric Brunet and Bernard Derrida, Probability distribution of the free energy of a directed polymer in a random medium, Physical Review E 2000, 61 (6), 6789-6801.
   arXiv, PRE, pdf.
   We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the n-th moment <Zⁿ> of the partition function is given by the ground state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder γ or in powers of n. For n small and n ~ (L γ)^-1/2, the moments <Zⁿ> take a scaling form which allows to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in 1+1 dimensions.
   P.A.C.S. numbers: 05.30, 05.70, 64.60 Cn.
5. Éric Brunet and Bernard Derrida, Effect of Microscopic Noise on Front Propagation, Journal of Statistical Physics 2001, 103 (1-2), 269-282.
   JSP, pdf.
   We study the effect of the noise due to microscopic fluctuations on the position of a one dimensional front propagating from a stable to an unstable region in the “linearly marginal stability case”. By simulating a very simple system for which the effective number N of particles can be as large as N=10¹⁵⁰, we measure the N dependence of the diffusion constant D_N of the front and the shift of its velocity v_N. Our results indicate that D_N ~ (log N)^-3. They also confirm our recent claim that the shift of velocity scales like v_min - v_N~ K(log N)^-2 and indicate that the numerical value of K is very close to the analytical expression K_approx obtained in our previous work using a simple cut-off approximation.
6. Éric Brunet, Fluctuations of the winding number of a directed polymer in a random medium , Physical Review E 2003, 68, 041101.
   arXiv, PRE, pdf.
   For a directed polymer in a random medium lying on an infinite cylinder, that is in 1+1 dimensions with finite width and periodic boundary conditions on the transverse direction, the winding number is simply the algebraic number of turns the polymer does around the cylinder. This paper presents exact expressions of the fluctuations of this winding number due to, first, the thermal noise of the system and, second, the different realizations of the disorder in the medium.
7. Éric Brunet and Bernard Derrida, Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium, Physical Review E 2004, 70, 016106.
   arXiv, PRE, pdf.
   We calculate exactly the velocity and diffusion constant of a microscopic stochastic model of N evolving particles which can be described by a noisy traveling wave equation with a noise of order N^-1/2. Our model can be viewed as the infinite range limit of a directed polymer in random medium with N sites in the transverse direction. Despite some peculiarities of the traveling wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoff approximation and to show that, in the weak noise limit, the position of the front can be completely described by the effect of the noise on the first particle.
8. Daniel ben-Avraham and Éric Brunet, On the relation between one-species diffusion-limited coalescence and annihilation in one dimension, Journal of Physics A 2005, 38, 3247-3252.
   arXiv, J. Phys. A, pdf.
   The close similarity between the hierarchies of multiple-point correlation functions for the diffusion-limited coalescence and annihilation processes has caused some recent confusion, raising doubts as to whether such hierarchies uniquely determine an infinite particle system. We elucidate the precise relations between the two processes, arriving at the conclusion that the hierarchy of correlation functions does provide a complete representation of a particle system on the line. We also introduce a new hierarchy of probability density functions for finding particles at specified locations and none in between. This hierarchy is computable for coalescence, through the method of empty intervals, and is naturally suited for questions concerning the ordering of particles on the line.
9. Éric Brunet, Bernard Derrida, Alfred H. Mueller and Stéphane Munier, A phenomenological theory giving the full statistics of the position of fluctuating pulled fronts, Physical Review E 2006, 73, 056126,
   arXiv, PRE, pdf.
   We propose a phenomenological description for the effect of a weak noise on the position of a front described by the Fisher-Kolmogorov-Petrovsky-Piscounov equation or any other travelling wave equation in the same class. Our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front. Our parameter-free analytical predictions for the velocity of the front, its diffusion constant and higher cumulants of its position agree with numerical simulations.
10. Éric Brunet, Bernard Derrida, Alfred H. Mueller and Stéphane Munier, Noisy traveling waves: effect of selection on genealogies, Europhysics Letters 2006, 76 (1), 1-7,
    arXiv, EPL, pdf.
    For a family of models of evolving population under selection, which can be described by noisy traveling-wave equations, the coalescence times along the genealogical tree scale like ln^α N, where N is the size of the population, in contrast with neutral models for which they scale like N. An argument relating this time scale to the diffusion constant of the noisy traveling wave leads to a prediction for α which agrees with our simulations. An exactly soluble case gives trees with statistics identical to those predicted for mean-field spin glasses by Parisi's theory.
11. Éric Brunet, Bernard Derrida, Alfred H. Mueller and Stéphane Munier, Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization , Physical Review E 2007, 76, 041104,
    arXiv, PRE, pdf.
    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.
12. Igor M. Rouzine, Éric Brunet and Claus O. Wilke, The traveling wave approach to asexual evolution: Muller's ratchet and speed of adaptation, Theoretical Population Biology 2008, 73, 24–46,
    arXiv, Theor. pop. biol., pdf.
    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.
13. Éric Brunet, Igor M. Rouzine and Claus O. Wilke, The stochastic edge in adaptive evolution, Genetics 2008, 179, 603–620,
    arXiv, Genetics, pdf.
    In a recent article, Desai and Fisher (2007) 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 τ until a new class of mutants has been established, and interpret 1/τ 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 τ: Desai and Fisher back-extrapolate the value of τ from the best-fit class' exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows non-exponentially 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 non-exponential back-extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al. 2003, 2007), and agree well with numerical simulations.
14. Éric Brunet, Bernard Derrida and Damien Simon, Universal tree structures in directed polymers and models of evolving populations , Physical Review E 2008, 78, 061102,
    arXiv, PRE, pdf.
    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.
15. Éric Brunet and Bernard Derrida, Statistics at the tip of a branching random walk and the delay of traveling waves, Europhysics Letters 2009, 87, 60010
    arXiv, Eur. Phys. L., pdf.
    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.
16. Éric Brunet, Bernard Derrida and Antoine Gerschenfeld, Fluctuations of the heat flux of a one-dimensional hard particle gas , Europhysics Letters 2010, 90, 20004
    arXiv, abstract, pdf.
    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).
17. Julien Berestycki, Éric Brunet, John W. Harris and Simon C. Harris, The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential , Statistics and Probability Letters 2010, 80, 1442-1446,
    arXiv, Stat. & Prob. L. pdf.
    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 βy², where β>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.
18. Éric Brunet and Bernard Derrida, A branching random walk seen from the tip, Journal of Statistical Physics 2011, 143, 420-446,
    arXiv, J. Stat. Phys., pdf.
    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.
19. Éric Brunet and Bernard Derrida, How genealogies are affected by the speed of evolution, Philosophical Magazine 2012, 92, 255-271,
    arXiv, Ph. Mag., pdf.
    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.
20. Elie Aïdékon, Julien Berestycki, Éric Brunet and Zhan Shi, Branching Brownian motion seen from its tip, Probab. Theory Relat. Fields, 2013, 157, 405-451
    arXiv, Probab. Theory Relat. Fields, pdf.
    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).
21. Éric Brunet and Bernard Derrida, Genealogies in simple models of evolution, J. Stat. Mech. 2013, P01006,
    arXiv, J. Stat. Mech., pdf.
    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, the time to reach the first common ancestors to the whole population and the time for all individuals to have all their ancestors in common are also logarithmic in the neutral case, as predicted by Chang []. We discuss how these times are modified in a simple way of introducing selection.
22. Julien Berestycki, Éric Brunet, John W. Harris, Simon C. Harris and Matthew I. Roberts, Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential, Stoch. Proc. Appl. 2015, 5, 2096-2145
    arXiv, Stoch. Proc. Appl., pdf.
    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∈[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.
23. Éric Brunet and Bernard Derrida, An exactly solvable travelling wave equation in the Fisher-KPP class, Journal of Statistical Physics 2015, 161 (4), 801-820.
    arXiv, J. Stat. Phys, pdf.
    For a simple one dimensional lattice version of a travelling wave equation, we obtain an exact relation between the initial condition and the position of the front at any later time. This exact relation takes the form of an inverse problem: given the times t_n at which the travelling wave reaches the positions n, one can deduce the initial profile. We show, by means of complex analysis, that a number of known properties of travelling wave equations in the Fisher-KPP class can be recovered, in particular Bramson's shifts of the positions. We also recover and generalize Ebert-van Saarloos' corrections depending on the initial condition.
24. Julien Berestycki, Éric Brunet and Zhan Shi, The number of accessible paths in the hypercube, Bernoulli 2016, 22 (2), 653-680
    arXiv, Bernoulli, pdf.
    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 ∈ [0,1]. We study the number θ of paths from the root (0,0,...,0) to the opposite corner (1,1,...,1) along which the values on the nodes form an increasing sequence. We show that if the value on the root is set to x=X/L then θ/L converges in law as L goes to infinity to exp(-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 ∈ [0,1]).
25. Julien Berestycki, Éric Brunet, A note of the convergence of the Fisher-KPP front centred around its α-level preprint 2016
    arXiv.
    We consider the solution u(x,t) of the Fisher-KPP equation ∂_tu=∂_x²u+u−u² centred around its α-level μ_t^(α) defined as u(μ_t^(α),t)=α. It is well known that for an initial datum that decreases fast enough, then u( μ_t^(α)+x,t) converges as t→∞ to the critical travelling wave. We study in this paper the speed of this convergence and the asymptotic expansion of μ_t^(α) for large t. It is known from Bramson that for initial conditions that decay fast enough, one has μ_t^(α)=2t−(3/2)ln t+Cste+o(1). Work is under way [] to show that the o(1) in the expansion is in fact a k^(α)/√t+O(t^ϵ−1) for any ϵ>0 for some k^(α), where it is not clear at this point whether k^(α) depends or not on α. We show that, unless the time derivative of μ_t^(α) has a very unexpected behaviour at infinity, the coefficient k^(α) does not, in fact, depend on α. We also conjecture that, for an initial condition that decays fast enough, one has in fact μ_t^(α)=2t−(3/2)ln t+Cste−(3π)/√t+g ln(t)/t+o(1/t) for some constant g which does not depend on α.
26. Julien Berestycki, Éric Brunet and Zhan Shi, Accessibility percolation with backsteps, ALEA 2017, 14, 45-62
    arXiv, ALEA, pdf.
    Consider a graph in which each site is endowed with a value called fitness. A path in the graph is said to be "open" or "accessible" if the fitness values along that path is strictly increasing. We say that there is accessibility percolation between two sites when such a path between them exists. Motivated by the so called House-of-Cards model from evolutionary biology, we consider this question on the L-hypercube {0,1}^L where the fitness values are independent random variables. We show that, in the large L limit, the probability that an accessible path exists from an arbitrary starting point to the (random) fittest site is no more than x*_1/2=1-1/2 sinh^-1(2)=0.27818... and we conjecture that this probability does converge to x*_1/2. More precisely, there is a phase transition on the value of the fitness x of the starting site: assuming that the fitnesses are uniform in [0,1], we show that, in the large L limit, there is almost surely no path to the fittest site if x>x*_1/2 and we conjecture that there are almost surely many paths if x<x*_1/2. If one conditions on the fittest site to be on the opposite corner of the starting site rather than being randomly chosen, the picture remains the same but with the critical point being now x*₁=1-sinh^-1(1)=0.11863.... Along the way, we obtain a large L estimation for the number of self-avoiding paths joining two opposite corners of the L-hypercube.
27. Julien Berestycki, Éric Brunet, Simon C. Harris and Matthew I. Roberts, Vanishing corrections for the position in a linear model of FKPP fronts Communications in Mathematical Physics 2017, 349, 857-893,
    arXiv, CIMP, pdf.
    Take the linearised FKPP equation ∂_t h=∂_xx h+h with boundary condition h(m(t),t)=0. Depending on the behaviour of the initial condition h₀(x)=h(x,0) we obtain the asymptotics — up to a o(1) term r(t) — of the absorbing boundary m(t) such that ω(x):=lim_t h(x +m(t),t) exists and is non-trivial. In particular, as in Bramson's results for the non-linear FKPP equation, we recover the celebrated −(3/2)log t correction for initial conditions decaying faster than x^νe^−x for some ν<−2. Furthermore, when we are in this regime, the main result of the present work is the identification (to first order) of the r(t) term which ensures the fastest convergence to ω(x). When h₀(x) decays faster than x^νe^−x for some ν<−3, we show that r(t) must be chosen to be −3(π/t)^1/2 which is precisely the term predicted heuristically by Ebert-van Saarloos in the non-linear case. When the initial condition decays as x^νe^−x for some ν∈[−3,−2), we show that even though we are still in the regime where Bramson's correction is −(3/2)log t, the Ebert-van Saarloos correction has to be modified. Similar results were recently obtained by Henderson using an analytical approach and only for compactly supported initial conditions.
28. Julien Berestycki, Éric Brunet, Simon C. Harris and Piotr Miłoś, Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift, Journal of Functional Analysis 2017, 273 (6), 2107-2143
    arXiv, JFA, pdf.
    We study a dyadic branching Brownian motion on the real line with absorption at 0, drift μ∈ℝ and started from a single particle at position x>0. When μ is large enough so that the process has a positive probability of survival, we consider K(t), the number of individuals absorbed at 0 by time t and for s≥0 the functions ω_s(x):=𝔼^x[s^K(∞)]. We show that ω_s<∞ if and only of s∈[0,s₀] for some s₀>1 and we study the properties of these functions. Furthermore, for s=0, ω(x):=ω₀(x)=ℙ^x(K(∞)=0) is the cumulative distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption. We give three descriptions of the family ω_s, s∈[0,s₀] through a single pair of functions, as the two extremal solutions of the Kolmogorov-Petrovskii-Piskunov (KPP) traveling wave equation on the half-line, through a martingale representation and as an explicit series expansion. We also obtain a precise result concerning the tail behavior of K(∞). In addition, in the regime where K(∞)>0 almost surely, we show that u(x,t):=ℙ^x(K(∞)=0) suitably centered converges to the KPP critical travelling wave on the whole real line.
29. Julien Berestycki, Éric Brunet and Bernard Derrida, Exact solution and precise asymptotics of a Fisher-KPP type front, Journal of Physics A 2017, 51 (3), 035204
    arXiv, JPA, pdf.
    The present work concerns a version of the Fisher-KPP equation where the nonlinear term is replaced by a saturation mechanism, yielding a free boundary problem with mixed conditions. Following an idea proposed in [BrunetDerrida.2015], we show that the Laplace transform of the initial condition is directly related to some functional of the front position μ_t. We then obtain precise asymptotics of the front position by means of singularity analysis. In particular, we recover the so-called Ebert and van Saarloos correction [EbertvanSaarloos.2000], we obtain an additional term of order log t / t in this expansion, and we give precise conditions on the initial condition for those terms to be present.
30. Julien Berestycki, Éric Brunet and Bernard Derrida, A new approach to computing the asymptotics of the position of Fisher-KPP fronts, Europhysics Letters 2018, 122 (1), 10001
    arXiv, EPL, pdf.
    This paper presents a novel way of computing front positions in Fisher-KPP equations. Our method is based on an exact relation between the Laplace transform of the initial condition and some integral functional of the front position. Using singularity analysis, one can obtain the asymptotics of the front position up to the O(log t/t) term. Our approach is robust and can be generalised to other front equations.
31. Julien Berestycki, Éric Brunet and Sarah Penington, Global existence for a free boundary problem of Fisher-KPP type, Nonlinearity 2019, 32, 3912
    arXiv, Nonlinearity, pdf.
    Motivated by the study of branching particle systems with selection, we establish global existence for the solution (u,μ) of the free boundary problem

    	∂tu =∂2xu+u		for t>0 and x>μt,
    	u(x,t)=1		for t>0 and x≤μt,
    	∂xu(μt,t)=0		for t>0,
    	u(x,0)=v(x)		for x∈R,

    when the initial condition v:R→[0,1] is non-increasing with v(x)→0 as x→∞ and v(x)→1 as x→−∞. We construct the solution as the limit of a sequence (u_n)_n≥1, where each u_n is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi et al. show that this global solution can be identified with the hydrodynamic limit of the so-called N-BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by killing the leftmost particle at each branching event.
32. Éric Brunet, Anh Dung Le, Alfred H. Mueller and Stéphane Munier, How to generate the tip of branching random walks evolved to large times, Europhysics Letters 2020, 131, 40002
    arXiv, EPL, pdf
    In a branching process, the number of particles increases exponentially with time, which makes numerical simulations for large times difficult. In many applications, however, only the region close to the extremal particles is relevant (the “tip”). We present a simple algorithm which allows to simulate a branching random walk in one dimension, keeping only the particles that arrive within some distance of the rightmost particle at a predefined time T . The complexity of the algorithm grows linearly with T . We can furthermore choose to require that the realizations have their rightmost particle arbitrarily far on the right from its typical position. We illustrate our algorithm by evaluating an observable for which no other practical method is known.
33. Julien Berestycki, Éric Brunet , James Nolen and Sarah Penington, A free boundary problem arising from branching Brownian motion with selection, Transactions of the American Mathematical Society 2021, 374, 6269-6329
    arXiv, Trans. AMS, pdf.
    We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in a companion paper. In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit.
34. Julien Berestycki, Éric Brunet, Cole Graham, Leonid Mytnik, Jean-Michel Roquejoffre and Lenya Ryzhik, The distance between the two BBM leaders, Nonlinearity 2022, 35(4), 1558.
    arXiv, Nonlinearity, pdf.
    We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d₁₂ of this random variable can be expressed in terms of PDEs related to the Fisher-KPP equation. We use such a representation to determine the sharp asymptotics of P( d₁₂>a) as a→+∞. These tail asymptotics were previously known to “exponential order”; we discover an algebraic correction to this behavior.
35. Julien Berestycki, Éric Brunet, James Nolen and Sarah Penington, Brownian bees in the infinite swarm limit , Annals of Probability 2022, 50(6), 2133-2177.
    arXiv, Ann. Probab., pdf.
    The Brownian bees model is a branching particle system with spatial selection. It is a system of N particles which move as independent Brownian motions in R^d and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that as N→∞ the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds, i.e. that as N→∞ the equilibrium density of the particle system converges to the steady state solution of the free boundary problem.
36. Julien Berestycki, Éric Brunet, Aser Cortines and Bastien Mallein, A simple backward construction of Branching Brownian motion with large displacement and applications Annales de l'IHP, probabilités et statistiques 2022, 58(4), 2094-2113.
    arXiv, Ann. IHP Probab. Statist., pdf.
    In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation probability of having a large displacement. As an application, we show that Bovier and Hartung (2015)'s results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein-Uhlenbeck processes.
37. Jess Benhabib, Éric Brunet and Mildred Hager, Innovation and imitation, Pure and Applied Functional Analysis 2022, 7(6), 2041-2063.
    arXiv, PAFA.
    We study several models of growth driven by innovation and imitation by a continuum of firms, focusing on the interaction between the two. We first investigate a model on a technology ladder where innovation and imitation combine to generate a balanced growth path (BGP) with compact support, and with productivity distributions for firms that are truncated power-laws. We start with a simple model where firms can adopt technologies of other firms with higher productivities according to exogenous probabilities. We then study the case where the adoption probabilities depend on the probability distribution of productivities at each time. We finally consider models with a finite number of firms, which by construction have firm productivity distributions with bounded support. Stochastic imitation and innovation can make the distance of the productivity frontier to the lowest productivity level fluctuate, and this distance can occasionally become large. Alternatively, if we fix the length of the support of the productivity distribution because firms too far from the frontier cannot survive, the number of firms can fluctuate randomly.
38. Camille Aron, Éric Brunet and Aditi Mitra, Traveling discontinuity at the quantum butterfly front, SciPost Phys. 2023 15, 042.
    arXiv, SciPost, pdf.
    We formulate a kinetic theory of quantum information scrambling in the context of a paradigmatic model of interacting electrons in the vicinity of a superconducting phase transition. We carefully derive a set of coupled partial differential equations that effectively govern the dynamics of information spreading in generic dimensions. They exhibit traveling wave solutions that are discontinuous at the boundary of the light cone, and have a perfectly causal structure where the solutions do not spill outside of the light cone.
39. Ankita Chakrabarti, Cyril Martins, Nicolas Laflorencie, Bertrand Georgeot, Éric Brunet and Gabriel Lemarié, Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree, SciPost Phys. 2023 15, 211
    arXiv, SciPost, pdf.
    We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant and algebraically vanishing boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.
40. Camille Aron, Éric Brunet and Aditi Mitra, Kinetics of information scrambling in correlated metals: Disorder-driven transition from shock wave to Fisher or Kolmogorov-Petrovsky-Piskunov dynamics, Phys. Rev. B 2023 108, L241106.
    arXiv, Phys. Rev. B, pdf.
    Quenched disorder slows down the scrambling of quantum information. Using a bottom-up approach, we formulate a kinetic theory of scrambling in a correlated metal near a superconducting transition, following the scrambling dynamics as the impurity scattering rate is increased. Within this framework, we rigorously show that the butterfly velocity v is bounded by the light-cone velocity v_lc set by the Fermi velocity. We analytically identify a disorder-driven dynamical transition occurring at small but finite disorder strength between a spreading of information characterized at late times by a discontinuous shock wave propagating at the maximum velocity v_lc, and a smooth traveling wave belonging to the Fisher or Kolmogorov-Petrovsky-Piskunov (FKPP) class and propagating at a slower, if not considerably slower, velocity v. In the diffusive regime, we establish the relation v²/𝜆_FKPP∼D_el, where 𝜆_FKPP is the Lyapunov exponent set by the inelastic scattering rate and D_el is the elastic diffusion constant.
41. Éric Brunet, Ahead of the Fisher-KPP front, Nonlinearity 2023, 36, 5541.
    arXiv, Nonlinearity, pdf.
    The solution h to the Fisher-KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h(c t, t) of the solution ahead of the front, at time t and position c t, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the non-linearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly explicit and irregular. The main tool of this paper is the so-called "magical expression" which relates the position of the front, the initial condition, and the quantity we investigate.

Phd thesis

Book

1. A. Aspect, F. Bouchet, É. Brunet, C. Cohen-Tannoudji, J. Dalibard, T. Damour, O. Darrigol, B. Derrida, P. Grangier, F. Laloë et J.-P. Pocholle, Einstein aujourd'hui, EDP sciences (2005)
   Ce livre a pour objectif de montrer combien les idées d'Einstein continuent à inspirer la science de ce début du XXIe siècle, après avoir révolutionné celle du XXe. Le livre contient sept contributions : une introduction historique et six articles retraçant les travaux les plus importants d'Einstein et leur impact sur la physique d'aujourd'hui : intrication de systèmes quantiques, condensation de Bose-Einstein, émission stimulée et laser, fluctuations et mouvement brownien, relativité générale, cosmologie.

Communications in conferences

• Systèmes aléatoires inhomogènes, Palaiseau, January 20–21, 1998.
• Statphys 20, Paris, July 20–24, 1998 (poster).
• Nordita Workshop on Nonequilibrium Physics, Copenhagen, September 23–25, 1999 (poster).
• Statistical Physics of Glassy and Non-Equilibrium Systems, Les Houches, January 24–29, 2000 (poster).
• Fronts, Fluctuations and Growth, Ann Arbor, May 19–24, 2002.
• Non-equilibrium Statistical Physics in Low Dimensions and Reaction-Diffusion Systems, Dresden, September 21–October 10, 2003 (poster).
• Geometry and Combinatorics in Physics, Moscow, March 2–7, 2004.
• La physique statistique hors équilibre et les systèmes désordonnés, Nancy, May 26–28, 2004.
• Workshop on Mathematical Population Genetics, Edinburgh, March 26–30, 2006.
• Inhomogeneous Random Systems, Paris, January 23–24, 2007.
• Points de vue sur les équations KPP, Marseilles, June 6–8 2007.
• Beg Rohu summer school of statistical physics and condensed matter, Beg Rohu, June 16–28, 2008.
• Disordered Systems; Spin Glasses, Montréal, June 8–13, 2009.
• Deterministic and Stochastic front propagation, Banff, March 21–26, 2010.
• Paris-Bath Brancing Structure Meeting, Bath, September 14–16, 2010.
• ECCS 2011, Frontiers in the Theory of Evolution, Vienna, September 12–16, 2011.
• Journée Darwin Paris, November 18, 2011.
• Disorder in Probability and Statistical Mechanics Modena, June 25–29, 2012.
• Extremes in branching random walk and branching Brownian motion Oberwolfach, April 21–27, 2013.
• Stochastic analysis: around the KPZ universality class Oberwolfach, June 1–7, 2014.
• Population dynamics and statistical physics in synergy Eindhoven, August 25–29, 2014.
• Front Propagation and Particle Systems Bannf, September 1–5, 2014.
• Probability and Biological Evolution Marseille, June 15-19, 2015.
• SPA 2015, 38^th Conference on Stochastic Processes and their Applications, Oxford, July 13-17, 2015 (invited session organizer).
• Bath-Beijing-Paris branching structures meeting, Beijing, May 14-18, 2018.
• Birs: Interacting particle systems and parabolic PDEs, Banff, August 26-31, 2018.