De nouveaux résultats sur la géométrie des mosaïques de Poisson-Voronoi et des mosaïques poissoniennes d'hyperplans. Etude du modèle de fissuration de Rényi-Widom

Abstract : This thesis deals with three models from stochastic geometry: the Poisson-Voronoi tessellation, Poisson hyperplane tessellation, and unidirectional crack model of Rényi-Widom. We first show the equivalence between the two historical methods for the statistical study of the tessellations: the convergence of the ergodic means and the Palm definition of the typical cell. We then provide in dimension two the law of the number of vertices of the typical cell and conditionally to this number, the laws of the positions of the bounds, the area and the perimeter. Moreover, we give the joint distribution of the radii of the largest (resp. smallest) disk centered at the origin containing (resp. contained in) the typical cell and we deduce the circular property of the ``large cells''. In the Poisson-Voronoi case, we connect the spectral function of the typical cell to the Brownian bridge, which provides the asymptotic behaviour of the law of the first eigenvalue in dimension two. For the Poissonian hyperplane tessellation, we use the Palm definition of the typical cell to deduce the explicit construction in any dimension of this cell by means of its in-ball and its circumscribed simplex. A rigorous proof of a result of R. E. Miles concerning the thickened hyperplane tessellations is also given. Besides, we model a cracking phenomenon by a unidimensional stationary process. We calculate the law of the typical inter-crack distance and we show that the consecutive points constitute an explicit conditioned renewal process.
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Pierre Calka. De nouveaux résultats sur la géométrie des mosaïques de Poisson-Voronoi et des mosaïques poissoniennes d'hyperplans. Etude du modèle de fissuration de Rényi-Widom. Mathématiques [math]. Université Claude Bernard - Lyon I, 2002. Français. ⟨tel-00448216⟩

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