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We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly…
When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample $n$ individuals from a population and trace…
We review the statistical properties of the genealogies of a few models of evolution. In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population in contrast with the linear growth of…
We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage, and are interested in the regime in which the…
Many population genetic models have been developed for the purpose of inferring population size and growth rates from random samples of genetic data. We examine two popular approaches to this problem, the coalescent and the…
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is…
Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with $N$ dormant individuals waking up independently of each other according to a given distribution. Once an individual…
Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K.…
The paper reviews the results obtained for spatial population models and the evolution of the genealogies of these populations during the last decade by the author and his coworkers. The focus is on their large scale behaviour and on the…
We investigate the evolution of populations of random Boolean networks under selection for robustness of the dynamics with respect to the perturbation of the state of a node. The fitness landscape contains a huge plateau of maximum fitness…
The value of a continuous character evolving on a phylogenetic tree is commonly modelled as the location of a particle moving under one-dimensional Brownian motion with constant rate. The Brownian motion model is best suited to characters…
We consider an exactly solvable model of branching random walk with random selection, which describes the evolution of a population with $N$ individuals on the real line. At each time step, every individual reproduces independently, and its…
First, we revisit the stochastic Luria-Delbr\"uck model: a classic two-type branching process which describes cell proliferation and mutation. We prove limit theorems and exact results for the mutation times, clone sizes, and number of…
Evolution occurs in populations of reproducing individuals. The structure of a biological population affects which traits evolve. Understanding evolutionary game dynamics in structured populations is difficult. Precise results have been…
Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing…
In a series of recent works it has been shown that a class of simple models of evolving populations under selection leads to genealogical trees whose statistics are given by the Bolthausen-Sznitman coalescent rather than by the well known…
We consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities $ \pm c $ and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study…
The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space-time, resampling the random walk…
We analyze the coalescing model where a `primary' colony grows and randomly emits secondary colonies that spread and eventually coalesce with it. This model describes population proliferation in theoretical ecology, tumor growth and is also…
The Brownian Web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in ${\mathbb R}\times{\mathbb R}$. We extend the earlier work of Arratia and of…