arXiv subjects

Olivier Hénard

Publications and source records attributed to Olivier Hénard.

12 recordsLinked to original sources

arXiv2026

Sharpness of the avalanche phase transition in the Bak--Sneppen model

We prove sharpness of the avalanche phase transition in the one-dimensional Bak--Sneppen model, for both the one-sided and isotropic update rules. In the one-sided model, the avalanche range is described by a nonlinear self-interacting jump process whose tails satisfy a triangular system. An extremal inequality for decreasing probability masses yields a polynomial lower bound on the range tail at the susceptibility threshold; this bound forces the appearance of infinite avalanches immediately above criticality and hence identifies the susceptibility and infinite-avalanche thresholds. A comparison of avalanche endpoints transfers this equality to the isotropic model. In both models, no infinite avalanche occurs at criticality, while the range has exponential moments at every strictly subcritical level. Combined with the range--duration and stationary-limit results of Meester and Znamenski, this identifies all three avalanche critical thresholds and proves the conjectured stationary product limit in both models.

math.PR↗

arXiv2025

Decompositions of CSBPs via Poissonian Intertwining

We revisit certain decompositions of continuous-state branching processes (CSBPs), commonly referred to as skeletal decompositions, through the lens of intertwining of semi-groups. Precisely, we associate to a CSBP $X$ with branching mechanism $ψ$ a family of $\mathbb{R}_+\times \mathbb{Z}_+$-valued branching processes $(X^λ, L^λ)$, indexed by a parameter $λ\in (0, \infty)$, that satisfies an intertwining relationship with $X$ through the Poisson kernel with parameter $λx$. The continuous component $X^λ$ has the same law as $X$, while the discrete component $L^λ$, conditionally on $X^λ_t$, has a Poisson distribution with parameter $λX^λ_t$. The law of $(X^λ, L^λ)$ depends on the position of $λ$ within $[0, \infty) = [0, ρ) \cup [ρ, \infty)$, where $ρ$ is the largest positive root of $ψ$. When $λ\geq ρ$, various well-known results concerning skeleton decompositions are recovered. In the supercritical case ($ρ> 0$), when $λ<ρ$, a novel phenomenon arises: a birth term appears in the skeleton, corresponding to a one-unit proportional immigration from the continuous to the discrete component. Along the way, the class of continuous-time branching processes taking values in $\mathbb{R}_+ \times \mathbb{Z}_+$ is constructed.

math.PR↗

arXiv2025

On cases where Litt's game is fair

A fair coin is flipped $n$ times, and two finite sequences of heads and tails (words) $A$ and $B$ of the same length are given. Each time the word $A$ appears in the sequence of coin flips, Alice gets a point, and each time the word $B$ appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt's game that recently set Twitter abuzz. We show that Litt's game is fair for any value of $n$ and any two words that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores; the fact that the inter-correlation does not come into play in this case may come up as a surprise.

math.CO↗

arXiv2024

A periodic Kingman model for the balance between mutation and selection

We consider a periodic extension of the classical Kingman non-linear model (Kingman, 1978) for the balance between selection and mutation in a large population. In the original model, the fitness distribution of the population is modeled by a probability measure on the unit interval evolving through a simple dynamical system in discrete time: selection acts through size-biasing, and the mutation probability and distribution are kept fixed through time. A natural extension of Kingman model is given by a periodic mutation environment; in this setting, we prove the convergence of the fitness distribution along subsequences and find an explicit criterion in terms of the Perron eigenvalue of an appropriately chosen matrix to decide whether an atom emerges at the largest fitness, a phenomenon usually called condensation.

math.PR↗

arXiv2024

The local limit of rooted directed animals on the square lattice

We consider the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and prove, in particular, that it consists of a single vertex at infinitely many (random) levels. Several martingales are found in connection with the confinement of the infinite directed animal on the non-negative coordinates.

math.PR↗

arXiv2022

Parking on the infinite binary tree

Let $(A_u : u \in \mathbb{B})$ be i.i.d.~non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $ \mathbb{B}$. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known that the parking process on $ \mathbb{B}$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We characterize those regimes in terms of the law of $A$ in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be "discontinuous".

math.PR↗

arXiv2022

The phase transition for parking on Galton--Watson trees

We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ<0$, then a positive fraction of the cars will not find a spot and exit the tree through the root. This confirms a conjecture of Goldschmidt and Przykucki.

math.PR↗

arXiv2018

On trees invariant under edge contraction

We study random trees which are invariant in law under the operation of contracting each edge independently with probability $p\in(0,1)$. We show that all such trees can be constructed through Poissonian sampling from a certain class of random measured $\R$-trees satisfying a natural scale invariance property. This has connections to exchangeable partially ordered sets, real-valued self-similar increasing processes and quasi-stationary distributions of Galton--Watson processes.

math.PR↗

arXiv2015

The fixation line in the $Λ$-coalescent

We define a Markov process in a forward population model with backward genealogy given by the $Λ$-coalescent. This Markov process, called the fixation line, is related to the block counting process through its hitting times. Two applications are discussed. The probability that the $n$-coalescent is deeper than the $(n-1)$-coalescent is studied. The distribution of the number of blocks in the last coalescence of the $n$-$\operatorname {Beta}(2-α,α)$-coalescent is proved to converge as $n\rightarrow\infty$, and the generating function of the limiting random variable is computed.

math.PR↗

arXiv2012

Stable continuous branching processes with immigration and Beta-Fleming-Viot processes with immigration

Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner et al. (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the alpha-stable continuous state branching process with the Beta(2-alpha, alpha)-generalized Fleming-Viot process. In a recent work, a new class of probability-measure valued processes, called M-generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called M-coalescents. The purpose of this article is to investigate the links between this new class of processes and the continuous-state branching processes with immigration. In the specific case of the $α$-stable branching process conditioned to be never extinct, we get that its genealogy is given, up to a random time change, by a Beta(2-alpha, alpha-1)-coalescent.

math.PR↗

arXiv2012

Change of measure in the lookdown particle system

We perform various changes of measure in the lookdown particle system of Donnelly and Kurtz. The first example is a product type h-transform related to conditioning a Generalized Fleming Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this h-transform by just "forgetting" some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright Fisher diffusion and explicit the associated pathwise decomposition. The second example, called the linear or additive h-transform, concerns a wider class of measure valued processes with spatial motion. Applications include: -a simple description of the additive h-transform of the Generalized Fleming Viot process, which confirms a suggestion of Overbeck for the usual Fleming Viot process -an immortal particle representation for the additive h-transform of the Dawson Watanabe process.

math.PR↗

arXiv2011

A Williams' Decomposition for Spatially Dependent Superprocesses

We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (William's decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diff usion and the superdiffusion are provided.

math.PR↗