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-Kolmogorov-Petrovsky-Piscounov equation, we evaluate the probability distribution P(c,t) of this empirical velocity c in the long-time t limit for c>2.

Exact solution of a Lévy walk model for anomalous heat transport.

The Lévy walk model is studied in the context of the anomalous heat conduction of one-dimensional systems. In this model, the heat carriers execute Lévy 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 Lévy walk model maintained in a steady state by contact with two heat baths (the open geometry).

Universal tree structures in directed polymers and models of evolving populations.

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.

Universal cumulants of the current in diffusive systems on a ring.

We calculate exactly the first cumulants of the integrated current and of the activity (which is the total number of changes of configurations) of the symmetric simple exclusion process on a ring with periodic boundary conditions. Our results indicate that for large system sizes the large deviation functions of the current and of the activity take a universal scaling form, with the same scaling function for both quantities. This scaling function can be understood either by an analysis of Bethe ansatz equations or in terms of a theory based on fluctuating hydrodynamics or on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim.

Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization.

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.

Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts.

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 traveling-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.

Distribution of current in nonequilibrium diffusive systems and phase transitions.

We consider diffusive lattice gases on a ring and analyze the stability of their density profiles conditionally to a current deviation. Depending on the current, one observes a phase transition between a regime where the density remains constant and another regime where the density becomes time dependent. Numerical data confirm this phase transition.

Phase transition in the ABC model.

Recent studies have shown that one-dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules. The ABC model, which comprises three particle species that diffuse asymmetrically around a ring, shows anomalous coarsening into a phase separated steady state. In the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one, no phase separation occurs and the steady state of the system is disordered.

Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium.

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.

Current fluctuations in nonequilibrium diffusive systems: an additivity principle.

We formulate a simple additivity principle allowing one to calculate the whole distribution of current fluctuations through a large one dimensional system in contact with two reservoirs at unequal densities from the knowledge of its first two cumulants. This distribution (which in general is non-Gaussian) satisfies the Gallavotti-Cohen symmetry and generalizes the one predicted recently for the symmetric simple exclusion process. The additivity principle can be used to study more complex diffusive networks including loops.

Exact free energy functional for a driven diffusive open stationary nonequilibrium system.

We obtain the exact probability exp[-LF([rho(x)])] of finding a macroscopic density profile rho(x) in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system L-->infinity. F, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, F is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in nonconvexity of F, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.

Sequence determination from overlapping fragments: a simple model of whole-genome shotgun sequencing.

Assembling fragments randomly sampled from along a sequence is the basis of whole-genome shotgun sequencing, a technique used to map the DNA of the human and other genomes. We calculate the probability that a random sequence can be recovered from a collection of overlapping fragments. We provide an exact solution for an infinite alphabet and in the case of constant overlaps.

Free energy functional for nonequilibrium systems: an exactly solvable case.

We consider the steady state of an open system in which there is a flux of matter between two reservoirs at different chemical potentials. For a large system of size N, the probability of any macroscopic density profile rho(x) is exp[-NF([rho])]; F thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for F is a nonlocal functional of rho, which yields the macroscopically long range correlations in the nonequilibrium steady state previously predicted by fluctuating hydrodynamics and observed experimentally.

Probability distribution of the free energy of a directed polymer in a random medium.

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 nth moment 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 gamma or in powers of n. For n small and n approximately (Lgamma)(-1/2), the moments take a scaling form which allows us to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer.

On the genealogy of a population of biparental individuals.

If one goes backward in time, the number of ancestors of an individual doubles at each generation. This exponential growth very quickly exceeds the population size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree.

Genetic distance and species formation in evolving populations.

We compare the behavior of the genetic distance between individuals in evolving populations for three stochastic models. In the first model reproduction is asexual and the distribution of genetic distances reflects the genealogical tree of the population. This distribution fluctuates greatly in time, even for very large populations.