arXiv subjects

Bastien Mallein

Publications and source records attributed to Bastien Mallein.

At least 19 recordsLinked to original sources

arXiv2026

Selection of the fittest or selection of the luckiest: the emergence of Goodhart's law in evolution

Biological evolution depends on the passing down to subsequent generations of genetic information encoding beneficial traits, and on the removal of unfit individuals by a selection mechanism. However, selection acts on phenotypes, and is affected by random contingencies. Thus, a combination of fitness and luck determines which individuals will successfully reproduce and give rise to the next generation. To understand how randomness in the selection mechanism affects the long-term patterns of evolution, we studied an idealized evolution model. We show through simulations and mathematical analysis, that the speed of adaptation increases with increasing selection pressure only up to a threshold. Beyond the threshold, any increase of the selection pressure results in more weight given to random effects rather than on genetic fitness in determining which individuals will successfully reproduce. This severely reduces the speed of adaptation and the diversity in the gene pool. Our findings may be considered as a biological instance of Goodhart's law: "When a measure becomes a target, it ceases to be a good measure". Finally, we show that this intricate response of evolution to natural selection can be mathematically explained by a novel phase transition for pulled traveling waves.

q-bio.PE↗

arXiv2026

The Derrida-Retaux model on a geometric Galton-Watson tree

We consider a generalized Derrida-Retaux model on a Galton-Watson tree with a geometric offspring distribution. For a class of recursive systems, including the Derrida-Retaux model with either a geometric or exponential initial distribution, we characterize the critical curve using an involution-type equation and prove that the free energy satisfies the Derrida-Retaux conjecture.

math.PR↗

arXiv2026

Extremal Process of Last Progeny Modified Branching Random Walks

We consider a last progeny modified branching random walk, in which the position of each particle at the last generation $n$ is modified by an i.i.d. copy of a random variable $Y$. Depending on the asymptotic properties of the tail of $Y$, we describe the asymptotic behaviour of the extremal process of this model as $n \to \infty$.

math.PR↗

arXiv2025

Elephants explore in spirals sometimes

We consider in this article an Elephant Random Walk evolving in the plane. Specifically, this is a reinforced stochastic process in which the $n$th step is given by a random rotation of one of the previous steps chosen uniformly at random. We obtain a central limit theorem for this process, which shows that the process follows a randomly rotated logarithmic spiral at large times, with Gaussian fluctuations.

math.PR↗

arXiv2025

Freezing in the Infinite-Bin Model

The infinite-bin model is a one-dimensional particle system on $\mathbb{Z}$ introduced by Foss and Konstantopoulos in relation with last passage percolation on complete directed acyclic graphs. In this model, at each integer time, a particle is selected at random according to its rank, and produces a child at the location immediately to its right. In this article, we consider the limiting distribution of particles after an infinite number of branching events have occurred. Under mild assumptions, we prove that the event (called freezing) that a location contains only a finite number of balls satisfies a $0-1$ law and we provide various criteria to determine whether freezing occurs.

math.PR↗

arXiv2025

Reinforced Galton--Watson processes III: Empirical offspring distributions

Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution depends on the empirical offspring distribution of each individual along its ancestral lineage. We are interested here in asymptotic properties of the empirical distributions observed in the population, such as concentration, evanescence and persistence. For this, we incorporate tools from the theory of large deviations to our preceding analysis [arXiv:2306.02476,arXiv:2310.19030].

math.PR↗

arXiv2024

The longest branches in a non-Markovian phylogenetic tree

Consider a Bellman--Harris-type branching process, in which individuals evolve independently of one another, giving birth after a random time $T$ to a random number $L$ of children. In this article, we study the asymptotic behaviour of the length of the longest branches of this branching process at time $t$, both pendant branches (corresponding to individuals still alive at time $t$) and interior branches (corresponding to individuals dead before time $t$).

math.PR↗

arXiv2024

The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift

Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-ρ$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten (1978) showed that this process dies out with probability one if and only if $ρ\geq \sqrt{2}$. We show that in the subcritical case when $ρ> \sqrt{2}$, the law of the process conditioned on survival until time $t$ converges as $t \rightarrow \infty$ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as $ρ\downarrow \sqrt{2}$ of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order $ε^{-1/3}$, and we obtain a limit result for the empirical distribution of the particle locations.

math.PR↗

arXiv2024

Reinforced Galton-Watson processes II: Large time behaviors

Reinforced Galton-Watson processes have been introduced in arxiv:2306.02476 as population models with non-overlapping generations, such that reproduction events along genealogical lines can be repeated at random. We investigate here some of their sample path properties such as asymptotic growth rates and survival, for which the effects of reinforcement on the evolution appear quite strikingly.

math.PR↗

arXiv2024

Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models

We consider directed random graphs, the prototype of which being the Barak-Erdős graph $\vec G(\mathbb Z, p)$, and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the graphs to certain particle systems that we call Infinite Bin Models (IBM). A number of limit theorems are shown. The goal of this paper is to present results along with techniques that have been used in this area. In the case of $\vec G(\mathbb Z, p)$ the last passage percolation constant $C(p)$ is studied in great detail. It is shown that $C(p)$ is analytic for $p>0$, has an interesting asymptotic expansion at $p=1$ and that $C(p)/p$ converges to $e$ like $1/(\log p)^2$ as $p \to 0$. The paper includes the study of IBMs as models on their own as well as their connections to stochastic models of branching processes in continuous or discrete time with selection. Several proofs herein are new or simplified versions of published ones. Regenerative techniques are used where possible, exhibiting random sets of vertices over which the graphs regenerate. When edges have random weights we show how the last passage percolation constants behave and when central limit theorems exist. When the underlying vertex set is partially ordered, new phenomena occur, e.g., there are relations with last passage Brownian percolation. We also look at weights that may possibly take negative values and study in detail some special cases that require combinatorial/graph theoretic techniques that exhibit some interesting non-differentiability properties of the last passage percolation constant. We also explain how to approach the problem of estimation of last passage percolation constants by means of perfect simulation.

math.PR↗

arXiv2024

Phase transition of the consistent maximal displacement of branching Brownian motion

Consider branching Brownian motion in which we begin with one particle at the origin, particles independently move according to Brownian motion, and particles split into two at rate one. It is well-known that the right-most particle at time $t$ will be near $\sqrt{2} t$. Roberts considered the so-called consistent maximal displacement and showed that with high probability, there will be a particle at time $t$ whose ancestors stayed within a distance $ct^{1/3}$ of the curve $s \mapsto \sqrt{2} s$ for all $s \in [0, t]$, where $c = (3 π^2)^{1/3}/\sqrt{2}$. We consider the question of how close the trajectory of a particle can stay to the curve $s \mapsto (\sqrt{2} + \varepsilon) s$ for all $s \in [0, t]$, where $\varepsilon> 0$ is small. We find that there is a phase transition, with the behavior changing when $t$ is of the order $\varepsilon^{-3/2}$. This result allows us to determine, for branching Brownian motion in which particles have a drift to the left of $\sqrt{2} + \varepsilon$ and are killed at the origin, the position at which a particle needs to begin at time zero for there to be a high probability that the process avoids extinction until time $t$.

math.PR↗

arXiv2023

Fragmentation processes and the convex hull of the Brownian motion in a disk

Motivated by the study of the convex hull of the trajectory of a Brownian motion in the unit disk reflected orthogonally at its boundary, we study inhomogeneous fragmentation processes in which particles of mass $m \in (0,1)$ split at a rate proportional to $|\log m|^{-1}$. These processes do not belong to the well-studied family of self-similar fragmentation processes. Our main results characterize the Laplace transform of the typical fragment of such a process, at any time, and its large time behavior. We connect this asymptotic behavior to the prediction obtained by physicists in \cite{DBBM22} for the growth of the perimeter of the convex hull of a Brownian motion in the disc reflected at its boundary. We also describe the large time asymptotic behavior of the whole fragmentation process. In order to implement our results, we make a detailed study of a time-changed subordinator, which may be of independent interest.

math.PR↗

arXiv2023

The extremal point process of branching Brownian motion in $\mathbb{R}^d$

We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $θ^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum δ_{θ^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\mathbb{S}^{d-1} \times \mathbb{R}$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(θ) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} θ$, where $D_\infty(θ)$ is the limit of the derivative martingale in direction $θ$ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasiński, Berestycki and Mallein (Ann. Inst. H. Poincaré 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).

math.PR↗

arXiv2023

Reinforced Galton-Watson processes I: Malthusian exponents

In a reinforced Galton-Watson process with reproduction law $\boldsymbolν$ and memory parameter $q\in(0,1)$, the number of children of a typical individual either, with probability $q$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability $1-q$, is given by an independent sample from $\boldsymbolν$. We estimate the average size of the population at a large generation, and in particular, we determine explicitly the Malthusian growth rate in terms of $\boldsymbolν$ and $q$. Our approach via the analysis of transport equations owns much to works by Flajolet and co-authors.

math.PR↗

arXiv2023

Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation

Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[-\infty,1]$ with essential supremum equal to $1$ (a charge of $-\infty$ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by $C(F)$. Even in the simplest case where $F=pδ_1 + (1-p)δ_{-\infty}$, corresponding to the longest path in the Barak-Erdős random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant $C(F)$. Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.

math.PR↗

arXiv2023

KPP traveling waves in the half-space

We study traveling waves of the KPP equation in the half-space with Dirichlet boundary conditions. We show that minimal-speed waves are unique up to translation and rotation but faster waves are not. We represent our waves as Laplace transforms of martingales associated to branching Brownian motion in the half-plane with killing on the boundary. We thereby identify the waves' asymptotic behavior and uncover a novel feature of the minimal-speed wave $Φ$. Far from the boundary, $Φ$ converges to a logarithmic shift of the 1D wave $w$ of the same speed: $\displaystyle \lim_{y \to \infty} Φ\big(x + \tfrac{1}{\sqrt{2}}\log y, y\big) = w(x)$.

math.AP↗

arXiv2022

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet $(σ^2,a,Λ)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(σ^2,a,Λ)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Lévy processes conditioned to stay positive.

math.PR↗

arXiv2022

On the length of the shortest path in a sparse Barak-Erdős graph

We consider an inhomogeneous version of the Barak-Erdős graph, i.e. a directed Erős-Rényi random graph on $\{1,\ldots,n\}$ with no loop. Given $f$ a Riemann-integrable non-negative function on $[0,1]^2$ and $γ> 0$, we define $G(n,f,γ)$ as the random graph with vertex set $\{1,\ldots,n\}$ such that for each $i < j$ the directed edge $(i,j)$ is present with probability $ p_{i, j}^{(n)} = \frac{f(i/n,j/n)}{n^γ}$, independently of any other edge. We denote by $L_n$ the length of the shortest path between vertices $1$ and $n$, and take interest in the asymptotic behaviour of $L_n$ as $n \to \infty$.

math.PR↗