arXiv subjects
Publications and source records attributed to Thierry Huillet.
In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.
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.
Consider the random Dirichlet partition of the interval into $n$ fragments with parameter $θ>0$. We recall the unordered Ewens sampling formulae from finite Dirichlet partitions. As this is a key variable for estimation purposes, focus is on the number of distinct visited species in the sampling process. These are illustrated in specific cases. We use these preliminary statistical results on frequencies distribution to address the following sampling problem: what is the estimated number of species when sampling is from Dirichlet populations? The obtained results are in accordance with the ones found in sampling theory from random proportions with Poisson-Dirichlet distribution. To conclude with, we apply the different estimators suggested to two different sets of real data.
We consider a random walk X_n in Z_+, starting at X_0=x>= 0, with transition probabilities P(X_{n+1}=X_n+1|X_n=y>=1)=1/2-δ/(4y+2δ) P(X_{n+1}=X_n+1|X_n=y>=1)=1/2+δ/(4y+2δ) and X_{n+1}=1 whenever X_n=0. We prove that the expectation value of X_n behaves like n^{1-(δ/2)} as n goes to infinity when δis in the range (1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.
The correlation function of a one-dimensional interface over a random substrate, bound to the substrate by a pressure term, is studied by Monte-Carlo simulation. It is found that the height correlation < h_i ; h_{i+j} >, averaged over the substrate disorder, fits a form exp(-(j/b)^c) to a surprising precision in the full range of j where the correlation is non-negligible. The exponent c increases from 1.0 to 1.5 when the interface tension is taken larger and larger.
Consider a list of $n$ files whose popularities are random. These files are updated according to the move-to-front rule and we consider the induced Markov chain at equilibrium. We give the exact limiting distribution of the search-cost per item as $n$ tends to infinity. Some examples are supplied.
In a probabilistic model of a film over a disordered substrate, Monte-Carlo simulations show that the film hangs from peaks of the substrate. The film profile is well approximated by a necklace of Wulff shapes. Such a necklace can be obtained as the infimum of a collection of Wulff shapes resting on the substrate. When the random substrate is given by iid heights with exponential distribution, we prove estimates on the probability density of the resulting peaks, at small density.
Suppose some random resource (energy, mass or space) $χ\geq 0$ is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume ${\Bbb E}χ=θ<\infty $ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can naturally be identified with the study of infinitely divisible random variables with Lévy measure concentrated on the interval% $.$ Special emphasis is put on these special partitioning models in the Poisson-Kingman class. The masses attached to the atoms of such partitions are sorted in decreasing order. Considering nearest- neighbors spacings yields a partition of unity which also deserves special interest. For such partition models, various statistical questions are addressed among which: correlation structure, cumulative energy of the first $K$ largest items, partition function, threshold and covering statistics, weighted partition, Rényi's, typical and size-biased fragments size. Several physical images are supplied. When the unbounded Lévy measure of $χ$ is $θx^{-1}\cdot {\bf I}% (x\in (0,1)) dx$, the spacings partition has Griffiths-Engen-McCloskey or GEM$(θ) $ distribution and $% χ$ follows Dickman distribution. The induced partition models have many remarkable peculiarities which are outlined. The case with finitely many (Poisson) fragments in the partition law is also briefly addressed. Here, the Lévy measure is bounded.
We compute the correlation function in the equilibrium version of Rényi's {\sl parking problem}. The correlation length is found to diverge as $2^{-1}π^{-2}(1-ρ)^{-2}$ when $ρ\nearrow1$ (maximum density) and as $π^{-2}(2ρ-1)^{-2}$ when $ρ\searrow1/2$ (minimum density).