arXiv subjects

Thierry Huillet

Publications and source records attributed to Thierry Huillet.

At least 19 recordsLinked to original sources

arXiv2023

Level numbers preserving transformations on excursionsdefined by random walks with state dependent jump laws

We study the number of individuals per level defined by excursions of random walks with state dependent jump law. These level numbersdetermine the probability of the excursion, and the set of transformations preserving the level numbers is, generically, the set of transformation that preserve the probability law of excursions.We compute the number of excursions having a fixed level numbers and we show that the class of shifts of excursions generate all the excursions having a fixed level numbers. We study the behaviorof the level numbers under the Vervaat transform and the Doob transform.

math.PR↗

arXiv2023

On Bernoulli trials with unequal harmonic success probabilities

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule-Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters.

math.PR↗

arXiv2023

On the balance between Emigration and Immigration as Random Walks on the non-negative integers

Life is on the razor's edge as resulting from competitive birth and death random forces. We illustrate this aphorism in the context of three Markov chain population models where systematic random immigration events promoting growth are simultaneously balanced with random emigration ones provoking thinning. The origin of mass removals are either determined by external demands or by aging leading to different conditions of stability.

physics.soc-ph↗

arXiv2023

Sterile versus prolific individuals pertaining to linear-fractional Bienaym{é}-Galton-Watson trees

In a Bienaymé-Galton-Watson process for which there is a positiveprobability for individuals of having no offspring, there is a subtlebalance and dependence between the sterile nodes (the dead nodes or leaves)and the prolific ones (the productive nodes) both at and up to the currentgeneration. We explore the many facets of this problem, especially in thecontext of an exactly solvable linear-fractional branching mechanism at allgeneration. Eased asymptotic issues are investigated. Relation of thisspecial branching process to skip-free to the left and simple random walks'excursions is then investigated. Mutual statistical information on theirshapes can be learnt from this association.

math.PR↗

arXiv2022

A Gibbsian random tree with nearest neighbour interaction

We revisit the random tree model with nearest-neighbour interaction as described in previous work, enhancing growth. When the underlying free Bienaymé-Galton-Watson (BGW) model is sub-critical, we show that the (non-Markov) model with interaction exhibits a phase transition between sub- and super-critical regimes. In the critical regime, using tools from dynamical systems, we show that the partition function of the model approaches a limit at rate $n^{-1}$ in the generation number $n$. In the critical regime with almost sure extinction, we also prove that the mean number of external nodes in the tree at generation $n$ decays like $n^{-2}$. Finally, we give a spin representation of the random tree, opening the way to tools from the theory of Gibbs states, including FKG inequalities. We extend the construction in previous work when the law of the branching mechanism of the free BGW process has unbounded support.

math.PR↗

arXiv2021

On decay-surge population models

We consider continuous space-time decay-surge population models which are semi- stochastic processes for which deterministically declining populations, bound to fade away, are rein- vigorated at random times by bursts or surges of random sizes. In a particular separable framework (in a sense made precise below) we provide explicit formulae for the scale (or harmonic) function and the speed measure of the process. The behavior of the scale function at infinity allows to formulate conditions under which such processes either explode or are transient at infinity, or Harris recurrent. A description of the structures of both the discrete-time embedded chain and extreme record chain of such continuous-time processes is supplied.

math.PR↗

arXiv2021

Asymptotic genealogies for a class of generalized Wright-Fisher models

We study a class of Cannings models with population size $N$ having a mixed multinomial offspring distribution with random success probabilities $W_1,\ldots,W_N$ induced by independent and identically distributed positive random variables $X_1,X_2,\ldots$ via $W_i:=X_i/S_N$, $i\in\{1,\ldots,N\}$, where $S_N:=X_1+\cdots+X_N$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into $N$ subintervals of lengths $W_1,\ldots,W_N$. Convergence results for the genealogy of these Cannings models are provided under regularly varying assumptions on the tail distribution of $X_1$. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet (2014) for the case when $X_1$ is Pareto distributed and complement those obtained by Schweinsberg (2003) for models where one samples without replacement from a supercritical branching process.

math.PR↗

arXiv2021

On population growth with catastrophes

In this paper we study a particular class of Piecewise deterministic Markov processes (PDMP's) which are semi-stochastic catastrophe versions of deterministic population growth models. In between successive jumps the process follows a flow describing deterministic population growth. Moreover, at random jump times, governed by state-dependent rates, the size of the population shrinks by a random amount of its current size, an event possibly leading to instantaneous local (or total) extinction. A special separable shrinkage transition kernel is investigated in more detail, including the case of total disasters. We discuss conditions under which such processes are recurrent (positive or null) or transient. To do so, we introduce a modified scale function which is used to compute, when relevant, the law of the height of excursions and to decide if the process is recurrent or not. The question of the finiteness of the time to extinction is investigated together with the evaluation of the mean time to extinction when the last one is finite. Some information on the embedded jump chain of the PDMP is also required when dealing with the classification of states 0 and infinity that we exhibit.

math.PR↗

arXiv2021

A generating function approach to Markov chains undergoing binomial catastrophes

In a Markov chain population model subject to catastrophes, random immigration events (birth), promoting growth, are in balance with the effect of binomial catastrophes that cause recurrent mass removal (death). Using a generating function approach, we study two versions of such population models when the binomial catastrophic events are of a slightly different random nature. In both cases, we describe the subtle balance between the two birth and death conflicting effects.

cond-mat.stat-mech↗

arXiv2021

Scaling features of two special Markov chains involving total disasters

Catastrophe Markov chain population models have received a lot of attention in the recent past. We herewith consider two special cases of such models involving total disasters, both in discrete and in continuous-time. Depending on the parameters range, the two models can show up a recurrence/transience transition and, in the critical case, a positive/null recurrence transition. The collapse transition probabilities are chosen in such a way that the models are exactly solvable and, in case of positive recurrence, intimately related to the extended Sibuya and Pareto-Zipf distributions whose divisibility and self-decomposability properties are shown relevant. The study includes: existence and shape of the invariant measure, time-reversal, return time to the origin, contact probability at the origin, extinction probability, height and length of the excursions, a renewal approach to the fraction of time spent in the catastrophic state, scale function, first time to collapse and first-passage times, divisibility properties.

math.PR↗

arXiv2020

Identities for Droplets with Circular Footprint on Tilted Surfaces

Exact mathematical identities are presented between the relevant parameters of droplets displaying circular contact boundary based on flat tilted surfaces. Two of the identities are derived from the force balance, and one from the torque balance. The tilt surfaces cover the full range of inclinations for sessile or pendant drops, including the intermediate case of droplets on a wall (vertical surface). The identities are put under test both by the available solutions of a linear response approximation at small Bond numbers as well as the ones obtained from numerical solutions, making use of the Surface Evolver software. The subtleties to obtain certain angle-averages appearing in identities by the numerical solutions are discussed in detail. It is argued how the identities are useful in two respects. First is to replace some unknown values in the Young-Laplace equation by their expressions obtained from the identities. Second is to use the identities to estimate the error for approximate analytical or numerical solutions without any reference to an exact solution.

physics.flu-dyn↗

arXiv2019

Revisiting John Lamperti's maximal branching process

Lamperti's maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.

math.PR↗

arXiv2019

Truncation in Duality and Intertwining Kernels

We study properties of truncations in the dual and intertwining process in the monotone case. The main properties are stated for the time-reversed process and the time of absorption of the truncated intertwining process.

math.PR↗

arXiv2018

Karlin-McGregor mutational occupancy problem revisited

Some population is made of n individuals that can be of p possible species (or types). The update of the species abundance occupancies is from a Moran mutational model designed by Karlin and McGregor in 1967. We first study the equilibrium species counts as a function of n, p and the total mutation probability $ν$ before considering various asymptotic regimes on n, p and $ν$. Running title: KMG Model with Mutations.

q-bio.PE↗

arXiv2017

Pinning of a drop by a junction on an incline

The shape of a drop pinned in a local equilibrium on an incline is a long-standing problem. The substrate can be homogeneous or heterogeneous and we herewith consider a drop pinned on an incline at the junction between a hydrophilic half-plane (the top half) and a hydropho-bic one (the bottom half). Relying on the equilibrium equations deriving from the balance of forces, we exhibit three scenarii depending on the way the contact line of the drop on the substrate either simply leans against the junction or overfills (partly or fully) the hydrophobic side. We draw some conclusions on the geometry of the overlap and the stability of these tentative equilibrium states. In the corresponding retention force factor, we find that a major role is played by the wetted length of the junction line. The predictions of the theory are illustrated and supported by extensive molecular dynamics simulations.

cond-mat.soft↗

arXiv2017

On a coalescence process and its branching genealogy

We define and analyze a coalescent process as a recursive box-filling process whose genealogy is given by an ancestral time-reversed, time-inhomogeneous Bienyamé-Galton-Watson process. Special interest is on the expected size of a typical box and its probability of being empty. Special cases leading to exact asymptotic computations are investigated when the coalescing mechanisms are either linear-fractional or quadratic.

math.PR↗

arXiv2017

On the genealogy and coalescence times of Bienaymé-Galton-Watson branching processes

Coalescence processes have received a lot of attention in the context of conditional branching processes with fixed population size and non-overlapping generations. Here we focus on similar problems in the context of the standard unconditional Bienaymé-Galton-Watson branching processes, either (sub)-critical or supercritical. Using an analytical tool, we derive the structure of some counting aspects of the ancestral genealogy of such processes, including: the transition matrix of the ancestral count process and an integral representation of various coalescence times distributions, such as the time to most recent common ancestor of a random sample of arbitrary size, including full size. We illustrate our results on two important examples of branching mechanisms displaying either finite or infinite reproduction mean, their main interest being to offer a closed form expression for their probability generating functions at all times. Large time behaviors are investigated.

math.PR↗

arXiv2017

Random evolutionary dynamics driven by fitness and house-of-cards mutations. Sampling formulae

We first revisit the multi-allelic mutation-fitness balance problem, especially when mutations obey a house of cards condition, where the discrete-time deterministic evolutionary dynamics of the allelic frequencies derives from a Shahshahani potential. We then consider multi-allelic Wright-Fisher stochastic models whose deviation to neutrality is from the Shahsha-hani mutation/selection potential. We next focus on the weak selection, weak mutation cases and, making use of a Gamma calculus, we compute the normalizing partition functions of the invariant probability densities appearing in their Wright-Fisher diffusive approximations. Using these results, Generalized Ewens sampling formulae (ESF) from the equilibrium distributions are derived. We start treating the ESF in the mixed mutation/selection potential case and then we restrict ourselves to the ESF in the simpler house-of-cards mutations only situation. We also address some issues concerning sampling problems from infinitely-many alleles weak limits.

q-bio.PE↗