Author
Jean-Baptiste Gouéré
Other affiliations: François Rabelais University, Claude Bernard University Lyon 1
Bio: Jean-Baptiste Gouéré is an academic researcher from University of Orléans. The author has contributed to research in topics: Percolation & Boolean model. The author has an hindex of 12, co-authored 45 publications receiving 655 citations. Previous affiliations of Jean-Baptiste Gouéré include François Rabelais University & Claude Bernard University Lyon 1.
Papers
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TL;DR: In this paper, the authors give topological and dynamical characterizations of mathematical quasicrystals, and prove that χ is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum.
Abstract: We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let denote the space of uniformly discrete subsets of the Euclidean space. Let denote the elements of that admit an autocorrelation measure. A Patterson set is an element of such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S ∈ is a Patterson set if and only if S is almost periodic in (
,
), where denotes the Besicovitch topology. Let χ be an ergodic random element of . We prove that χ is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets.
97 citations
TL;DR: In this paper, the authors consider a class of branching-selection particle systems, similar to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in the velocity of a front due to a cutoff".
Abstract: We consider a class of branching-selection particle systems on \({\mathbb{R}}\) similar to the one considered by E. Brunet and B. Derrida in their 1997 paper “Shift in the velocity of a front due to a cutoff”. Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size N of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate (log N)−2. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of N independent branching random walks killed below a linear space-time barrier.
79 citations
TL;DR: In this article, the authors considered the Poisson Boolean model of percolation and showed that there is a subcritical phase if and only if E(R d ) is finite, where R denotes the radius of the balls around Poisson points and d denotes the dimension
Abstract: We consider the Poisson Boolean model of continuum percolation We show that there is a subcritical phase if and only if E(R d ) is finite, where R denotes the radius of the balls around Poisson points and d denotes the dimension We also give related results concerning the integrability of the diameter of subcritical clusters
64 citations
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TL;DR: In this article, the authors consider the Poisson Boolean model of percolation and show that for any positive real number n, the integrability of the point process is integrable if and only if the mean volume of the radii of the balls is finite.
Abstract: We consider the so-called Poisson Boolean model of continuum percolation. At each point of an homogeneous Poisson point process on the Euclidean space $\R^d$, we center a ball with random radius. We assume that the radii of the balls are independent, identically distributed and independent of the point process. We denote by $\Sigma$ the union of the balls and by $S$ the connected component of $\Sigma$ that contains the origin. We show that $S$ is almost surely bounded for small enough density $\lambda$ of the point process if and only if the mean volume of the balls is finite. Let us denote by $D$ the diameter of $S$ and by $R$ one of the random radii. We also show that, for all positive real number $s$, $D^s$ is integrable for small enough $\lambda$ if and only if $R^{d+s}$ is integrable.
60 citations
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TL;DR: In this article, the authors introduced a topology on the space of uniformly discrete subsets of the Euclidean space, where the autocorrelation measure is known to exist in the context of ergodic point processes.
Abstract: We introduce a topology ${\cal T}$ on the space $U$ of uniformly discrete subsets of the Euclidean space. Assume that $S$ in $U$ admits a unique autocorrelation measure. The diffraction measure of $S$ is purely atomic if and only if $S$ is almost periodic in $(U,{\cal T})$. This result relates idealized quasicrystals to almost periodicity. In the context of ergodic point processes, the autocorrelation measure is known to exist. Then, the diffraction measure is purely atomic if and only if the dynamical system has a pure point spectrum. As an illustration, we study deformed model sets.
54 citations
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TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
1,031 citations
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TL;DR: Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves as discussed by the authors, and it has been applied to population genetics and other fields such as spin glass models.
Abstract: Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
163 citations
156 citations
TL;DR: In this paper, the authors consider topological dynamical systems arising from locally compact Abelian groups on compact spaces of translation bounded measures and show that such a system has a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
Abstract: Certain topological dynamical systems are considered that arise from actions of $\sigma$-compact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
131 citations
TL;DR: In this paper, the authors consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles.
Abstract: We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order (logN) 3 , in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu’s continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the non-rigorous predictions by Brunet, Derrida, Muller, and Munier for a closely related model.
121 citations