The aim of this PhD thesis is to prove some properties related to Delaunay triangulations on point processes with random conductances that allow to verify the validity of some assumptions in the theorems contained in [2,3,4]. As a consequence of these theorems, one can obtain the stochastic homogenization of the random walk on the Delaunay triangulation, the hydrodynamic limit of the simple exclusion process on the Delaunay triangulation and the scaling limit of the directional conductivity of the random resistor networks on Delaunay triangulation. We start by introducing the random environment. According to [6], we introduce the Voronoi tessellation and we discuss some geometrical properties. We further give the definition of points in general quadratic position and we present a criterion that guarantees the boundedness of the Voronoi cells. We have that boundedness of Voronoi cells is a sufficient condition for Voronoi cells to be convex polytopes ([5]). Then we give a brief introduction of the theory of point processes and simple point processes referring to [1]. First, we provide the basic definitions of random measures, counting measures (simple counting measures) and point processes (simple point processes). We present the stationary random measures and the properties related to the stationary point processes (stationary simple point processes). Then we provide the definition of Campbell measure and of local Palm distributions of a random measure. As a special case, we consider the local Palm distributions of a stationary random measures ξ with finite intensity. Called M the first moment measure of ξ, we obtain that the family of local Palm distributions coincides M −a.e. with the Palm distribution. Finally we discuss an ergodic theorem when ξ is a stationary ergodic point process with finite density. Later, we argue sufficient conditions assuring that the points of a simple point process are in general position or in reinforced general position almost surely. The definition of points in reinforced general position, introduced in [8], coincides with that of points in general quadratic position. Following [8], we give a characterization of the simple point processes that are in general position, or general reinforced position, in terms of the reduced k−th moment measure. As an application of the previous results, we present the cases of Poisson point processes and Gibbsian point processes. Afterwards, we present the exclusion process on Z^d. Following [7], we construct the exclusion process on Z^d and we discuss some of its properties. We also review the proof of the hydrodynamic limit of the symmetric exclusion process on Z^d. Finally, we present the original contribution of the PhD work which was realised in collaboration with A.Faggionato. We consider the Voronoi tessellation associated to an ergodic and stationary point process on R^d. We introduce the Delaunay triangulation as the graph with vertex set given by the point process and with edges between vertexes whose Voronoi cells share a (d-1)-dimensional face. We also attach to each edge a random conductance. On the Delaunay triangulation we consider the random walk with random conductances, the simple exclusion process obtained by multiple random walks as above under hardcore interaction and the resistor network with random conductances. With the purpose of applying the general results obtained in [2,3,4] in the case of Delaunay triangulations, we investigate the validity of suitable moment bounds and we look for a bond percolation result for the existence of the simple exclusion process. We give specific criteria in our context assuring the validity of the assumptions made in [2,3,4]. As said at the beginning, as a byproduct between our results and [2,3,4], one can obtain the homogenization of the massive Poisson equation associated to the random walk, the hydrodynamic limit in path space of the simple exclusion process and the scaling limit of the directional conductivity of the resistor network. [1] D.J. Daley, D. Vere-Jones; “An introduction to the theory of point processes.” New York, Springer Verlag, 1988. [2] A. Faggionato; "Stochastic homogenization of random walks on point processes." Ann. Inst. H. Poincaré Probab. Statist. (to appear). arXiv:2009.08258 (2020). [3] A. Faggionato; "Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization." arXiv:2011.11361 (2020). [4] A. Faggionato; "Scaling limit of the conductivity of random resistor networks on simple point processes." arXiv:2108.11258 (2021). [5] D. Hug, W. Weil; “Lectures on convex geometry.” Graduate Texts in Mathematics 286, Springer International Publishing, 2020. [6] J. Møller; “Lectures on random Voronoi tessellation.” Lecture Notes in Statistics 87, Springer Verlag, New York, 1994. [7] T. Seppäläinen; “Translation invariant exclusion processes.” Book in progress, https://people.math.wisc.edu/~seppalai/excl-book/ajo.pdf. [8] H. Zessin; “Point processes in general position.” Izvestiya NAN Armenii Matematika 1, 75–82 (2008); preprinted in Journal of Contemporary Mathematical Analysis 43 (1), 59—65 (2008).

Homogenization, simple exclusion processes and random resistor networks on Delaunay triangulations

TAGLIAFERRI, CRISTINA
2022-07-07

Abstract

The aim of this PhD thesis is to prove some properties related to Delaunay triangulations on point processes with random conductances that allow to verify the validity of some assumptions in the theorems contained in [2,3,4]. As a consequence of these theorems, one can obtain the stochastic homogenization of the random walk on the Delaunay triangulation, the hydrodynamic limit of the simple exclusion process on the Delaunay triangulation and the scaling limit of the directional conductivity of the random resistor networks on Delaunay triangulation. We start by introducing the random environment. According to [6], we introduce the Voronoi tessellation and we discuss some geometrical properties. We further give the definition of points in general quadratic position and we present a criterion that guarantees the boundedness of the Voronoi cells. We have that boundedness of Voronoi cells is a sufficient condition for Voronoi cells to be convex polytopes ([5]). Then we give a brief introduction of the theory of point processes and simple point processes referring to [1]. First, we provide the basic definitions of random measures, counting measures (simple counting measures) and point processes (simple point processes). We present the stationary random measures and the properties related to the stationary point processes (stationary simple point processes). Then we provide the definition of Campbell measure and of local Palm distributions of a random measure. As a special case, we consider the local Palm distributions of a stationary random measures ξ with finite intensity. Called M the first moment measure of ξ, we obtain that the family of local Palm distributions coincides M −a.e. with the Palm distribution. Finally we discuss an ergodic theorem when ξ is a stationary ergodic point process with finite density. Later, we argue sufficient conditions assuring that the points of a simple point process are in general position or in reinforced general position almost surely. The definition of points in reinforced general position, introduced in [8], coincides with that of points in general quadratic position. Following [8], we give a characterization of the simple point processes that are in general position, or general reinforced position, in terms of the reduced k−th moment measure. As an application of the previous results, we present the cases of Poisson point processes and Gibbsian point processes. Afterwards, we present the exclusion process on Z^d. Following [7], we construct the exclusion process on Z^d and we discuss some of its properties. We also review the proof of the hydrodynamic limit of the symmetric exclusion process on Z^d. Finally, we present the original contribution of the PhD work which was realised in collaboration with A.Faggionato. We consider the Voronoi tessellation associated to an ergodic and stationary point process on R^d. We introduce the Delaunay triangulation as the graph with vertex set given by the point process and with edges between vertexes whose Voronoi cells share a (d-1)-dimensional face. We also attach to each edge a random conductance. On the Delaunay triangulation we consider the random walk with random conductances, the simple exclusion process obtained by multiple random walks as above under hardcore interaction and the resistor network with random conductances. With the purpose of applying the general results obtained in [2,3,4] in the case of Delaunay triangulations, we investigate the validity of suitable moment bounds and we look for a bond percolation result for the existence of the simple exclusion process. We give specific criteria in our context assuring the validity of the assumptions made in [2,3,4]. As said at the beginning, as a byproduct between our results and [2,3,4], one can obtain the homogenization of the massive Poisson equation associated to the random walk, the hydrodynamic limit in path space of the simple exclusion process and the scaling limit of the directional conductivity of the resistor network. [1] D.J. Daley, D. Vere-Jones; “An introduction to the theory of point processes.” New York, Springer Verlag, 1988. [2] A. Faggionato; "Stochastic homogenization of random walks on point processes." Ann. Inst. H. Poincaré Probab. Statist. (to appear). arXiv:2009.08258 (2020). [3] A. Faggionato; "Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization." arXiv:2011.11361 (2020). [4] A. Faggionato; "Scaling limit of the conductivity of random resistor networks on simple point processes." arXiv:2108.11258 (2021). [5] D. Hug, W. Weil; “Lectures on convex geometry.” Graduate Texts in Mathematics 286, Springer International Publishing, 2020. [6] J. Møller; “Lectures on random Voronoi tessellation.” Lecture Notes in Statistics 87, Springer Verlag, New York, 1994. [7] T. Seppäläinen; “Translation invariant exclusion processes.” Book in progress, https://people.math.wisc.edu/~seppalai/excl-book/ajo.pdf. [8] H. Zessin; “Point processes in general position.” Izvestiya NAN Armenii Matematika 1, 75–82 (2008); preprinted in Journal of Contemporary Mathematical Analysis 43 (1), 59—65 (2008).
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