Publications and Preprints
- On a two-parameter Yule-Simon distribution.
With Jean Bertoin. Preprint (2020).
[ abstract, arXiv ]We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in Bertoin [B19] in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in the mentioned paper. Finally, by superposing mutations to the branching process, we propose a model which leads to the full two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon [S55] on limiting word frequencies.
- On a class of random walks with reinforced memory.
[ abstract, arXiv ]This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Schütz and Trimper [ST04] with stronger reinforcement mechanisms, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability p. With probability 1-p, the random walk performs a step independent of the past. The weight of the remembered step is increased by an additive factor b≥0, making it likelier to repeat the step again in the future. A combination of techniques from the theory of urns, branching processes and α-stable processes enables us to discuss the limit behavior of reinforced versions of both the Elephant Random Walk and its α-stable counterpart, the so-called Shark Random Swim introduced by Businger [B18]. We establish phase transitions, separating subcritical from supercritical regimes.
- Classification of scaling limits of uniform
quadrangulations with a boundary.
With Grégory Miermont and Gourab Ray. Ann. Probab. 6(47) (2019), 3397–3477.
[ abstract, arXiv, published version ]We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter θ and the infinite-volume Brownian disk of perimeter σ. We also obtain various coupling and limit results clarifying the relation between these objects.
- Uniform infinite half-planar quadrangulations with skewness.
With Loïc Richier. Electron. J. Probab. 23(54) (2018), 1–43.
[ abstract, arXiv, published version ]We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness (UIHPQp for short, with p∈[0,1/2] measuring the skewness). They interpolate between Kesten's tree corresponding to p=0 and the usual UIHPQ corresponding to p=1/2. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way. Our main result shows that the family (UIHPQp)p approximates the Brownian half-planes BHPθ, θ≥0, recently introduced in Baur, Miermont and Ray [BMR16]. For p<1/2, we give a description of the UIHPQp in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.
- Weak limits for the largest subpopulations in Yule
processes with high mutation probabilities.
With Jean Bertoin. Adv. Appl. Prob. 49(3) (2017), 877–902.
[ abstract, arXiv, published version ]We consider a Yule process until the total population reaches size n >>1, and assume that neutral mutations occur with high probability 1-p (in the sense that each child is a new mutant with probability 1-p, independently of the other children), where p=pn<<1. We establish a general strategy for obtaining Poisson limit laws and law of large numbers for the number of subpopulations exceeding a given size and apply it to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter pn tending to zero.
- Geodesic rays in the uniform infinite half-planar quadrangulation
return to the boundary.
With Grégory Miermont and Loïc Richier. Lat. Am. J. Probab. Math. Stat. 13 (2016), 1123–1149.
[ abstract, arXiv, published version ]We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien in [CC15] by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.
- Elephant Random Walks and their connection to Pólya-type urns.
With Jean Bertoin. Phys. Rev. E 94 052134 (2016).
[ abstract, arXiv, published version ]In this note, we explain the connection between the Elephant Random Walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced by Schütz and Trimper in [ST04] to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a parameter p between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of p. Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.
- Sur la destruction de grands arbres aléatoires récursifs.
With Jean Bertoin. Gazette des Mathématiciens 150 (2016), 41–47.
[ abstract, hal, published version ]L'objet de ce texte est de présenter aussi simplement que possible les principaux résultats d'un programme récent portant sur l'étude des différents régimes observés lors de la destruction d'un arbre aléatoire uniforme en supprimant ses arêtes les une après les autres, uniformément au hasard.
- An invariance principle for a class of non-ballistic random walks in
Probab. Theory Relat. Fields. 166(1) (2016), 463–514.
[ abstract, arXiv, published version ]We are concerned with random walks on Zd, d≥3, in an i.i.d. random environment with transition probabilities ε-close to those of simple random walk. We assume that the environment is balanced in one fixed coordinate direction, and invariant under reflection in the coordinate hyperplanes. The invariance condition was used in [BB14] as a weaker replacement of isotropy to study exit distributions. We obtain precise results on mean sojourn times in large balls and prove a quenched invariance principle, showing that for almost all environments, the random walk converges under diffusive rescaling to a Brownian motion with a deterministic (diagonal) diffusion matrix. We also give a concrete description of the diffusion matrix. Our work extends the results of Lawler [L82], where it is assumed that the environment is balanced in all coordinate directions.
- Percolation on random recursive trees.
Random Struct. Algor. 48(4) (2016), 655–680.
[ abstract, arXiv, published version ]We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree-indexed process of cluster sizes to the genealogical tree of a continuous-state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [Be14]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous-time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation.
- Exit laws from large balls of (an)isotropic random walks in random
With Erwin Bolthausen. Ann. Probab. 43(6) (2015), 2859–2948.
[ abstract, arXiv, published version ]We study exit laws from large balls in Zd, d≥3, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. Our work includes the results on isotropic random walks in random environment of Bolthausen and Zeitouni [BZ07]. Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a somewhat different approach was given in the first author's thesis [B13]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environment.
- The fragmentation process of an infinite recursive tree and
Ornstein-Uhlenbeck type processes.
With Jean Bertoin. Electron. J. Probab. 20(98) (2015), 1–20.
[ abstract, arXiv, published version ]We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition Π(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process Π is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of Π(t). We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.
- Cutting edges at random in large recursive trees.
With Jean Bertoin. Stochastic Analysis and Applications 2014, Springer Proceedings in Mathematics & Statistics 100, 51–76.
[ abstract, arXiv, published version ]We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT's and present recent results on the cluster sizes in the supercritical regime.
- Long-time behavior of random walks in random environment.
Part of the PhD thesis (2013), 1–92.
[ abstract, arXiv ]We study behavior in space and time of random walks in an i.i.d. random environment on Zd, d≥3. It is assumed that the measure governing the environment is isotropic and concentrated on environments that are small perturbations of the fixed environment corresponding to simple random walk. We develop a revised and extended version of the paper of Bolthausen and Zeitouni [BZ07] on exit laws from large balls, which, as we hope, is easier to follow. Further, we study mean sojourn times in balls. This work is part of the author's PhD thesis under the supervision of Erwin Bolthausen.
- On a ternary coalescent process.
Lat. Am. J. Probab. Math. Stat. X(2) (2013), 561–589.
[ abstract, arXiv (extended version), published version ]We present a coalescent process where three particles merge at each coagulation step. Using a random walk representation, we prove duality with a fragmentation process, whose fragmentation law we specify explicitly. Furthermore, we give a second construction of the coalescent in terms of random binary forests and study asymptotic properties. Starting from N particles of unit mass, we obtain under an appropriate rescaling when N tends to infinity a well-known binary coalescent, the so-called standard additive coalescent.
- Long-time behavior of random walks in random environment and asymptotics of a ternary coalescent process. PhD thesis, University of Zurich (2013).
- Metastabilität von reversiblen Diffusionsprozessen. Diploma thesis, University of Bonn (2009).