Bastien Mallein
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Bastien Mallein has a total of 48 co-authors. Here are the top co-authors by number of shared publications:
- Julien Berestycki
Department of Statistics and Magdalen College, University of Oxford, Oxford, UK
4 shared publications Alexander Iksanov
Operations Research Department, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
3 shared publicationsAser Cortines
University of Zürich
3 shared publicationsJean Bertoin
Institute of Mathematics University of Zurich Zurich Switzerland
3 shared publications- Sanjay Ramassamy
Institut de Physique Théorique Université Paris-Saclay CNRS CEA France
3 shared publications - Xinxin Chen
School of Mathematical Sciences, Beijing Normal University, Beijing, China
3 shared publications
Latest Publications
- Open Access
- 6 June 2026
- Published by Walter de Gruyter GmbH in Stochastics and Quality Control
- Vol. 41 (2) , 97-113
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 𝑛th 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.
- 6 June 2026
- Published by Walter de Gruyter GmbH in Stochastics and Quality Control
- Vol. 41 (2) , 97-113
Open AccessElephants 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 𝑛th 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.
- 1 April 2026
- Published by Agence Bibliographique de l'Enseignement Supérieur
Branching random walks, time-inhomogeneous environment, selection
Marches aléatoires branchantes, temps inhomogène, sélection: On s'intéresse dans cette thèse au modèle de la marche aléatoire branchante, un système de particules qui évoluent au court du temps en se déplaçant et se reproduisant de façon indépendante. Le but est d'étudier le rythme auquel ces particules se déplacent, dans deux variantes particulières de marches aléatoires branchantes. Dans la première variante, la façon dont les individus se déplacent et se reproduisent dépend du temps. Ce modèle a été introduit par Fang et Zeitouni en 2010. Nous nous intéresserons à trois types de dépendance en temps : une brusque modification du mécanisme de reproduction des individus après un temps long ; une lente évolution de ce mécanisme à une échelle macroscopique ; et des fluctuations aléatoires à chaque génération. Dans la seconde variante, le mécanisme de reproduction est constant, mais les individus subissent un processus de sélection darwinien. La position d'un individu est interprétée comme son degré d'adaptation au milieu, et le déplacement d'un enfant par rapport à son parent représente l'héritage des gènes. Dans un tel processus, la taille maximale de la population est fixée à une certaine constante N, et à chaque étape, seuls les N plus à droite sont conservés. Ce modèle a été introduit par Brunet, Derrida, Mueller et Munier, et étudié par Bérard et Gouéré en 2010. Nous nous sommes intéressés dans un premier temps à une variante de ce modèle, qui autorise quelques grands sauts. Dans un second temps, nous avons considéré que la taille totale N de la population dépend du temps.
- 1 April 2026
- Published by Agence Bibliographique de l'Enseignement Supérieur
Branching random walks, time-inhomogeneous environment, selection
Marches aléatoires branchantes, temps inhomogène, sélection: On s'intéresse dans cette thèse au modèle de la marche aléatoire branchante, un système de particules qui évoluent au court du temps en se déplaçant et se reproduisant de façon indépendante. Le but est d'étudier le rythme auquel ces particules se déplacent, dans deux variantes particulières de marches aléatoires branchantes. Dans la première variante, la façon dont les individus se déplacent et se reproduisent dépend du temps. Ce modèle a été introduit par Fang et Zeitouni en 2010. Nous nous intéresserons à trois types de dépendance en temps : une brusque modification du mécanisme de reproduction des individus après un temps long ; une lente évolution de ce mécanisme à une échelle macroscopique ; et des fluctuations aléatoires à chaque génération. Dans la seconde variante, le mécanisme de reproduction est constant, mais les individus subissent un processus de sélection darwinien. La position d'un individu est interprétée comme son degré d'adaptation au milieu, et le déplacement d'un enfant par rapport à son parent représente l'héritage des gènes. Dans un tel processus, la taille maximale de la population est fixée à une certaine constante N, et à chaque étape, seuls les N plus à droite sont conservés. Ce modèle a été introduit par Brunet, Derrida, Mueller et Munier, et étudié par Bérard et Gouéré en 2010. Nous nous sommes intéressés dans un premier temps à une variante de ce modèle, qui autorise quelques grands sauts. Dans un second temps, nous avons considéré que la taille totale N de la population dépend du temps.
- Open Access
- 22 April 2024
- Published by Wiley in Random Structures & Algorithms
- Vol. 65 (2) , 387-410
Reinforced Galton–Watson processes I: Malthusian exponents
- 22 April 2024
- Published by Wiley in Random Structures & Algorithms
- Vol. 65 (2) , 387-410
Open AccessReinforced Galton–Watson processes I: Malthusian exponents
- Open AccessThe 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 $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (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(\theta) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} \theta $, where $D_\infty(\theta)$ is the limit of the derivative martingale in direction $\theta$ 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 Stasi\'nski, Berestycki and Mallein (Ann. Inst. H. Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).Open AccessThe 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 $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (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(\theta) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} \theta $, where $D_\infty(\theta)$ is the limit of the derivative martingale in direction $\theta$ 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 Stasi\'nski, Berestycki and Mallein (Ann. Inst. H. Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).
- Open Access
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 $F=p\delta_1 + (1-p)\delta_{-\infty}$, corresponding to the longest path in the Barak-Erd\H{o}s 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.Open AccessEstimation 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 $F=p\delta_1 + (1-p)\delta_{-\infty}$, corresponding to the longest path in the Barak-Erd\H{o}s 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. - Open Access
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: $\displaystyle \lim_{y \to \infty} Φ\big(x + \tfrac{1}{\sqrt{2}}\log y, y\big) = w(x)$.Open AccessKPP 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: $\displaystyle \lim_{y \to \infty} Φ\big(x + \tfrac{1}{\sqrt{2}}\log y, y\big) = w(x)$. - 1 January 2023
- Published by Institute for Applied and Pure Mathematics (IMPA) in Latin American Journal of Probability and Mathematical Statistics
- Vol. 20 (2) , 905-940
Branching Brownian motion conditioned on small maximum
- 1 January 2023
- Published by Institute for Applied and Pure Mathematics (IMPA) in Latin American Journal of Probability and Mathematical Statistics
- Vol. 20 (2) , 905-940
Branching Brownian motion conditioned on small maximum
- Open Access
- 16 June 2022
- Published by EDP Sciences in ESAIM: Probability and Statistics
- Vol. 26, 283-303
A tractable non-adaptative group testing method for non-binary measurements
The original problem of group testing consists in the identification of defective items in a collection, by applying tests on groups of items that detect the presence of at least one defective element in the group. The aim is then to identify all defective items of the collection with as few tests as possible. This problem is relevant in several fields, among which biology and computer sciences. In the present article we consider that the tests applied to groups of items returns aload, measuring how defective the most defective item of the group is. In this setting, we propose a simple non-adaptative algorithm allowing the detection of all defective items of the collection. Items are put on ann×ngrid and pools are organised as lines, columns and diagonals of this grid. This method improves on classical group testing algorithms using only the binary response of the test. Group testing recently gained attraction as a potential tool to solve a shortage of COVID-19 test kits, in particular for RT-qPCR. These tests return the viral load of the sample and the viral load varies greatly among individuals. Therefore our model presents some of the key features of this problem. We aim at using the extra piece of information that represents the viral load to construct a one-stage pool testing algorithm on this idealized version. We show that under the right conditions, the total number of tests needed to detect contaminated samples can be drastically diminished.
- 16 June 2022
- Published by EDP Sciences in ESAIM: Probability and Statistics
- Vol. 26, 283-303
Open AccessA tractable non-adaptative group testing method for non-binary measurements
The original problem of group testing consists in the identification of defective items in a collection, by applying tests on groups of items that detect the presence of at least one defective element in the group. The aim is then to identify all defective items of the collection with as few tests as possible. This problem is relevant in several fields, among which biology and computer sciences. In the present article we consider that the tests applied to groups of items returns aload, measuring how defective the most defective item of the group is. In this setting, we propose a simple non-adaptative algorithm allowing the detection of all defective items of the collection. Items are put on ann×ngrid and pools are organised as lines, columns and diagonals of this grid. This method improves on classical group testing algorithms using only the binary response of the test. Group testing recently gained attraction as a potential tool to solve a shortage of COVID-19 test kits, in particular for RT-qPCR. These tests return the viral load of the sample and the viral load varies greatly among individuals. Therefore our model presents some of the key features of this problem. We aim at using the extra piece of information that represents the viral load to construct a one-stage pool testing algorithm on this idealized version. We show that under the right conditions, the total number of tests needed to detect contaminated samples can be drastically diminished.
- Open Access
On the length of the shortest path in a sparse Barak-Erdős graph
We consider an inhomogeneous version of the Barak-Erd\H{o}s graph, i.e. a directed Er\H{o}s-R\'enyi random graph on $\{1,\ldots,n\}$ with no loop. Given $f$ a Riemann-integrable non-negative function on $[0,1]^2$ and $\gamma > 0$, we define $G(n,f,\gamma)$ 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^\gamma}$, 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 $n \to \infty$.Open AccessOn the length of the shortest path in a sparse Barak-Erdős graph
We consider an inhomogeneous version of the Barak-Erd\H{o}s graph, i.e. a directed Er\H{o}s-R\'enyi random graph on $\{1,\ldots,n\}$ with no loop. Given $f$ a Riemann-integrable non-negative function on $[0,1]^2$ and $\gamma > 0$, we define $G(n,f,\gamma)$ 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^\gamma}$, 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 $n \to \infty$. - 9 January 2022
- Published by Springer Nature in Journal of Theoretical Probability
- Vol. 35 (4) , 2569-2599
A Simple Method to Find All Solutions to the Functional Equation of the Smoothing Transform
- 9 January 2022
- Published by Springer Nature in Journal of Theoretical Probability
- Vol. 35 (4) , 2569-2599
A Simple Method to Find All Solutions to the Functional Equation of the Smoothing Transform