Thierry Huillet

RANDOM COVERING OF THE CIRCLE: THE SIZE OF THE CONNECTED COMPONENTS

Consider a circle of circumference 1. Throw n points at random onto this circle and append to each of these points a clockwise arc of length s. The resulting random set is a union of a random number of connected components, each with specific size. Using tools designed by Steutel, we compute the joint distribution of the lengths of the connected components. Asymptotic results are presented when n goes to ∞ and s to 0 jointly according to different regimes.

A Duality Approach to the Genealogies of Discrete Nonneutral Wright-Fisher Models

Discrete ancestral problems arising in population genetics are investigated. In the neutral case, the duality concept has proved of particular interest in the understanding of backward in time ancestral process from the forward in time branching population dynamics. We show that duality formulae still are of great use when considering discrete non-neutral Wright-Fisher models. This concerns a large class of non-neutral models with completely monotone (CM) bias probabilities. We show that most classical bias probabilities used in the genetics literature fall within this CM class or are amenable to it through some `reciprocal mechanism which we define. Next, using elementary algebra on CM functions, some suggested novel evolutionary mechanisms of potential interest are introduced and discussed.

Energy cascades as branching processes with emphasis on Neveu's approach to Derrida's random energy model

Continuous-space-time branching processes (CSBP) are investigated in order to model random energy cascades. CSBPs are based on spectrally positive Lévy processes and, as such, are characterized by their corresponding Laplace exponents. Special emphasis is put on the CSBPs of Feller, Lamperti and Neveu and on their Poisson point process representations. The Neveu model (either supercritical or subcritical) is of particular interest in physics for its connection with the random energy model of Derrida, as revisited by Ruelle. Exploiting some connections between the partition functions of energy and the Poisson-Dirichlet distributions of Pitman and Yor, some information on the zero-temperature limit is extracted. Finally, for the subcritical versions of the three models, we compute the distribution of some of their interesting features: extinction time and probability, area under the profile (total energy) and width (maximal energy).

ON L EVY STABLE AND SEMISTABLE DISTRIBUTIONS

This work emphasizes the special role played by semistable and log-semistable distributions as relevant statistical models of various observable and internal variables in physics. Besides of their representation, some of their remarkable properties (chiefly semi-self-similarity) are displayed in some detail. One of their characteristic features is a log-periodic variation of the scale parameter which appears in the standard L evy -stable distributions whose Fourier representations are re-derived in a self-contained way.

ON THE WAITING TIME PARADOX AND RELATED TOPICS

Consider a pure recurrent positive renewal process generated by some interarrival waiting time. The waiting time paradox reveals that, asymptotically, the time interval covering ones arrival in the file is statistically longer than the typical waiting time. Special properties are known to hold, were the waiting time to be infinitely divisible, two particular subclasses of interest being the exponential power mixtures and the Levys ones. These models are revisited in some detail. Questions related to these problems are investigated and special examples of interest are underlined.

Is superhydrophobicity robust with respect to disorder

We consider theoretically the Cassie-Baxter and Wenzel states describing the wetting contact angles for rough substrates. More precisely, we consider different types of periodic geometries such as square protrusions and disks in 2D, grooves and nanoparticles in 3D and derive explicitly the contact angle formulas. We also show how to introduce the concept of surface disorder within the problem and, inspired by biomimetism, study its effect on superhydrophobicity. Our results, quite generally, prove that introducing disorder, at fixed given roughness, will lower the contact angle: a disordered substrate will have a lower contact angle than a corresponding periodic substrate. We also show that there are some choices of disorder for which the loss of superhydrophobicity can be made small, making superhydrophobicity robust.Graphical abstract

On a deposition process on the circle with disorder

Throw n points sequentially and at random onto a unit circle and append a clockwise arc (or rod) of length s to each such point. The resulting random set (the free gas of rods) is a union of a random number of clusters with random sizes modelling a free deposition process on a one-dimensional substrate. A variant of this model is investigated in order to take into account the role of the disorder, θ > 0; this involves Dirichlet(θ) distributions. For such free deposition processes with disorder θ, we shall be interested in the occurrence times and probabilities, as n grows, of two specific types of configurations: those avoiding overlapping rods (the hard-rod gas) and those for which the largest gap is smaller than the rod length s (the packing gas). Special attention is paid to the thermodynamic limit when ns = ρ for some finite density ρ of points. The occurrence of parking configurations, those for which hard-rod and packing constraints are both fulfilled, is then studied. Finally, some aspects of these problems are investigated in the low-disorder limit θ ↓ 0 as n ↑ ∞ while nθ = γ > 0. Here, Poisson-Dirichlet(γ) partitions play some role.

Reversing the drift of the Ehrenfest urn model and three conditionings

We consider a drift-reversed version of the celebrated Ehrenfest urn process with N balls. For this dual process, the boundaries are assumed to be absorbing and so the killing times at the boundaries play a central role. Three natural conditionings on the fixation/extinction events pertaining to this model are investigated. Some spectral information on the conditioned Markoff chains is obtained, allowing us to draw precise new conclusions on their limiting behaviors.

Fluctuations Analysis of Finite Discrete Birth and Death Chains with Emphasis on Moran Models with Mutations

The Moran model is a discrete-time birth and death Markov chain describing the evolution of the number of type 1 alleles in a haploid population with two alleles whose total size is preserved during the course of evolution. Bias mechanisms such as mutations or selection can affect its neutral dynamics. For the ergodic Moran model with mutations, we get interested in the fixation probabilities of a mutant, the growth rate of fluctuations, the first hitting time of the equilibrium state starting from state , the first return time to the equilibrium state, and the first hitting time of starting from , together with the time needed for the walker to reach its invariant measure, again starting from . For the last point, an appeal to the notion of Siegmund duality is necessary, and a cutoff phenomenon will be made explicit. We are interested in these problems in the large population size limit . The Moran model with mutations includes the heat exchange models of Ehrenfest and Bernoulli-Laplace as particular cases; these were studied from the point of view of the controversy concerning irreversibility (-theorem) and the recurrence of states.

On Discrete-Time Multiallelic Evolutionary Dynamics Driven by Selection

We revisit some problems arising in the context of multiallelic discrete-time evolutionary dynamics driven by fitness. We consider both the deterministic and the stochastic setups and for the latter both the Wright-Fisher and the Moran approaches. In the deterministic formulation, we construct a Markov process whose Master equation identifies with the nonlinear ndeterministic evolutionary equation. Then, we draw the attention on a class of fitness matrices that plays some role in the important matter of polymorphism: the class of strictly ultrametric fitness matrices. In the random cases, we focus on fixation probabilities, on various conditionings on nonfixation, and on (quasi)stationary distributions.