# Rencontre de Géométrie Stochastique

## Mercredi 17 et Jeudi 18 Octobre 2018

Organisateurs : Nicolas Chenavier, David Coupier, David Dereudre, Viet Chi Tran

### Programme

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# Programme

 Mercredi 17 octobre 2018 Salle de séminaire, au 3ème étage du Bâtiment M3 14:00-15:00 Marie Théret: Some subcritical properties in continuum percolation We consider the Boolean model S: at each point of a homogeneous Poisson point process on the d-dimensional Euclidean space, we center a ball of random radius, and we denote by S the union of the balls. We consider the connected component of the origin in S, and look at some of its properties: is it bounded ? what is the integrability of its volume, of its diameter, or of the number of balls of S it contains ? We will try to understand how these properties are connected, in particular when the radii of the balls are not bounded. This is a joint work with Jean-Baptiste Gouéré (Institut Denis Poisson, Tours). 15:15-16:15 Jean-François Marckert: The combinatorics of the colliding bullets The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let (Vi)1≤ i≤ n be an i.i.d. family of random variables with uniform distribution on [0,1]; shoot n bullets one after another at times 1,2,..., n, where the ith bullet has speed Vi. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalisation of this model. While the distribution is relatively simple, the proof is surprisingly intricate and mixes combinatorial and geometric arguments (Common work with Nicolas Broutin). 17:00-18:00 Cyrille Lucas: Uniform IDLA IDLA is a growth model in which a random set is built recursively. New particules started at the origin perform a random walk until they settle on previously unoccupied sites. In our uniform version, new particules are started according to the uniform distribution on occupied sites. We will see that this self-dependance modifies the behaviour of the set but not its limiting shape, which remains the Euclidean ball. An alternative construction of the aggregate using the genealogical tree of the particules is particularly useful for bounding the outer error. Jeudi 18 octobre 2018 Salle Duhem, au premier étage du Bâtiment M3 9:00-10:00 Thomas Bonis: Random neighborhood and graphs Laplacians in machine learning In order to perform non-linear data analysis, one popular approach consists in computing a graph from the data and using a graph analysis algorithm such as spectral clustering or Laplacian eigenmaps (dimensionality reduction). Such graphs are usually obtained by taking the data points as vertices and by putting an edge between two data points if they are sufficiently close from one another. Understanding the behaviour of graph anlaysis algorithms on these neighborhood graphs is thus key to provide insight regarding their uses in machine learning application. In this talk, I will focus on graph analysis algorithms which rely on graph Laplacians. After describing the properties of these Laplacians and how they are used in specific algorithms, I will discuss their consistency in the light of recent results proving the convergence of graph Laplacians of random geometric graphs. Finally, I will present some open problems and potential generalizations regarding the use of graph Laplacians in machine learning. 10:30-11:30 Christian Hirsch: Graph-based Pólya urns on countable networks We examine a reinforcement model on the edges of a countably infinite network. The edge weights evolve according to a stochastic process subject to the following dynamics. First, vertices fire at times given by a Poisson point process. Second, if a vertex fires at a point in time, then one of the incident edges is selected with a probability proportional to the current weight to a power a strictly smaller than 1. Third, the weight of the selected edge is increased by 1. We analyze the percolation properties of the family of edges used a positive proportion of time. Our methods apply both to discrete deterministic graphs such as the lattice as well as to random geometric graphs. This talk is based on joint work with Mark Holmes and Victor Kleptsyn. 11:30-12:30 Youri Davydov: Remarks on asymptotic independence The next points will be in discussion: How to choose the definition of asymptotic independence? Criters for asymptotic independence. Case of tight sequences. Strong asymptotic independence and its relations with conditions of weak dependence.