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Bernard Derrida

Publications and source records attributed to Bernard Derrida.

At least 37 records · Page 2Linked to original sources
arXiv2018

A new approach to computing the asymptotics of the position of Fisher-KPP fronts

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.

cond-mat.stat-mech↗
arXiv2017

Finite size corrections to the Parisi overlap function in the GREM

We investigate the effects of finite size corrections on the overlap probabilities in the Generalized Random Energy Model (GREM) in two situations where replica symmetry is broken in the thermodynamic limit. Our calculations do not use replicas, but shed some light on what the replica method should give for finite size corrections. In the gradual freezing situation, which is known to exhibit full replica symmetry breaking, we show that the finite size corrections lead to a modification of the simple relations between the sample averages of the overlaps $ Y_k $ between $ k $ configurations predicted by replica theory. This can be interpreted as fluctuations in the replica block size with a \emph{negative} variance. The mechanism is similar to the one we found recently in the random energy model [1]. We also consider a simultaneous freezing situation, which is known to exhibit one step replica symmetry breaking. We show that finite size corrections lead to full replica symmetry breaking and give a more complete derivation of the results presented in [2] for the directed polymer on a tree.

cond-mat.dis-nn↗
arXiv2017

Exact solution and precise asymptotics of a Fisher-KPP type front

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.

cond-mat.stat-mech↗
arXiv2017

Slower deviations of the branching Brownian motion and of branching random walks

We have shown recently how to calculate the large deviation function of the position $X_{\max}(t) $ of the right most particle of a branching Brownian motion at time $t$. This large deviation function exhibits a phase transition at a certain negative velocity. Here we extend this result to more general branching random walks and show that the probability distribution of $X_{\max}(t)$ has, asymptotically in time, a prefactor characterized by non trivial power law.

math-ph↗
arXiv2016

Correlations of the density and of the current in non-equilibrium diffusive systems

We use fluctuating hydrodynamics to analyze the dynamical properties in the non-equilibrium steady state of a diffusive system coupled with reservoirs. We derive the two-time correlations of the density and of the current in the hydrodynamic limit in terms of the diffusivity and the mobility. Within this hydrodynamic framework we discuss a generalization of the fluctuation dissipation relation in a non-equilibrium steady state where the response function is expressed in terms of the two-time correlations. We compare our results to an exact solution of the symmetric exclusion process. This exact solution also allows one to directly verify the fluctuating hydrodynamics equation.

cond-mat.stat-mech↗
arXiv2016

On the genealogy of branching random walks and of directed polymers

It is well known that the mean field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.

cond-mat.stat-mech↗
arXiv2016

Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

Consider a one-dimensional branching Brownian motion, and rescale the coordinate and time so that the rates of branching and diffusion are both equal to $1$. If $X_1(t)$ is the position of the rightmost particle of the branching Brownian motion at time $t$, the empirical velocity $c$ of this rightmost particle is defined as $c=X_1(t)/t$. Using the Fisher-KPP equation, we evaluate the probability distribution ${\mathcal P(c,t)}$ of this empirical velocity $c$ in the long time $t$ limit for $c > 2$. It was already known that, for a single seed particle, ${\mathcal P(c,t)} \sim \exp \,[-(c^2/4-1)t]$ up to a prefactor that can depend on $c$ and $t$. Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated to other traveling wave equations.

cond-mat.stat-mech↗
arXiv2016

Large deviations for the branching Brownian motion in presence of selection or coalescence

The large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk (BRW), or in a branching Brownian motion (BBM). More recently a number of generalizations of the BBM and of the BRW have been considered where selection or coalescence mechanisms tend to limit the exponential growth of the number of particles. Here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations: the $L$-BBM, the $N$-BBM, and the CBRW (coalescing branching random walk) which is closely related to the noisy FKPP equation. Our approach allows us to obtain only upper bounds on these large deviation functions. One noticeable feature of our results is their non analytic dependence on the parameters (such as the coalescence rate in the CBRW).

math-ph↗
arXiv2015

An exactly solvable travelling wave equation in the Fisher-KPP class

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.

cond-mat.stat-mech↗
arXiv2015

Large deviation function of a tracer position in single file diffusion

Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.

cond-mat.stat-mech↗
arXiv2014

Finite size corrections in the random energy model and the replica approach

We present a systematic and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as "complex replica numbers". Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the "complex replica numbers" near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally, our approach allows one to see why some apparently diverging series or integrals are harmless.

cond-mat.dis-nn↗
arXiv2014

The depinning transition in presence of disorder: a toy model

We introduce a toy model, which represents a simplified version of the problem of the depinning transition in the limit of strong disorder. This toy model can be formulated as a simple renormalization transformation for the probability distribution of a single real variable. For this toy model, {the critical line is known exactly in one particular case} and it can be calculated perturbatively in the general case. One can also show that, at the transition, there is no strong disorder fixed distribution accessible by renormalization. Instead, both our numerical and analytic approaches indicate a transition of infinite order (of the Berezinskii-Kosterlitz-Thouless type). We give numerical evidence that this infinite order transition persists for the problem of the depinning transition with disorder on the hierarchical lattice.

cond-mat.stat-mech↗
arXiv2013

Universal current fluctuations in the symmetric exclusion process and other diffusive systems

We show, using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs are the same on an arbitrary large finite domain in dimension $d$ as in the one dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.

cond-mat.stat-mech↗
arXiv2013

Log-periodic critical amplitudes: a perturbative approach

Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.

cond-mat.stat-mech↗
arXiv2013

Exact solution of a Levy walk model for anomalous heat transport

The Levy walk model is studied in the context of the anomalous heat conduction of one dimensional systems. In this model the heat carriers execute Levy-walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations and the temperature profile of the Levy-walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is non-locally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size dependent cut-off time is necessary for the Levy walk model to behave as mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes.

cond-mat.stat-mech↗
arXiv2012

Genealogies in simple models of evolution

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.

q-bio.PE↗
arXiv2011

Anomalous long-range correlations at a non-equilibrium phase transition

Non-equilibrium diffusive systems are known to exhibit long-range correlations, which decay like the inverse 1/L of the system size L in one dimension. Here, taking the example of the ABC model, we show that this size dependence becomes anomalous (the decay becomes a non-integer power of L) when the diffusive system approaches a second-order phase transition. This power-law decay as well as the L-dependence of the time-time correlations can be understood in terms of the dynamics of the amplitude of the first Fourier mode of the particle densities. This amplitude evolves according to a Langevin equation in a quartic potential, which was introduced in a previous work to explain the anomalous behavior of the cumulants of the current near this second-order phase transition. Here we also compute some of the cumulants away from the transition and show that they become singular as the transition is approached.

cond-mat.stat-mech↗