Marches au hasard sur des graphes géométriques aléatoires engendrés par des processus ponctuels

Abstract : Random walks on random graphs embedded in Rd appear naturally in problems arisingfrom statistical mechanics such that the description of flows, molecules or heat diffusionsin random and irregular environments. The general idea is to extend known results forrandom walks on Zd or on random perturbations of the grid to results for random walkson graphs generated by point processes in Rd.In this thesis, we consider nearest neighbor random walks on graphs depending on thegeometry of a random infinite locally finite set of points. More precisely, given a realisationof a simple stationary point process in Rd, a connected infinite and locally finite graph G isconstructed. This graph is then possibly equipped with a conductance function C, that is apositive function defined on its edge set. Examples of graphs studied in this manuscript arethe Delaunay triangulation, the Gabriel graph, the creek-crossing graphs and the skeletonof the Voronoi tiling generated by the point process. We study properties of the simplerandom walk or of a random walk associated with the conductance C on such graphs.The main results concern the characterisation of the recurrence or transience of therandom walks and the description of their diffusive scaling limits. Under suitable assumptionson the underlying point process and the conductance function, we show that therandom walks on the Delaunay triangulation, the Gabriel graph and the skeleton of theVoronoi tiling generated by almost every realisation of the point process are recurrent ifd = 2 and transient if d 3. We state an annealed invariance principle for simple randomwalks starting from the origin on the Delaunay triangulation, the Gabriel graph and thecreek-crossing graphs generated by Palm measures of suitable point processes. Finally,we show a quenched invariance principle for simple random walks on random Delaunaytriangulations.This thesis uses tools from both stochastic geometry (point processes, Palm measures,random graphs ...) and the theory of random walks (links with electrical networks theory,the environment seen from the particle,...).
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Arnaud Rousselle. Marches au hasard sur des graphes géométriques aléatoires engendrés par des processus ponctuels. Probabilités [math.PR]. Université de Rouen, 2014. Français. ⟨tel-01096364⟩

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