The Ginibre point process is one of the main examples of determinantal point processes on the complex plane. It forms a recurring model in stochastic matrix theory as well as in practical applications. Since its introduction in random matrix theory, the Ginibre point process has also been used to model random phenomena where repulsion is observed. In this paper, we modify the classical Ginibre point process in order to obtain a determinantal point process more suited for simulation. We also compare three different methods of simulation and discuss the most efficient one depending on the application at hand.Mathematics Subject Classification: 60G55, 65C20.
Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.
Abstract-In this paper, we aim at reducing power consumption in wireless sensor networks by turning off supernumerary sensors. Random simplicial complexes are tools from algebraic topology which provide an accurate and tractable representation of the topology of wireless sensor networks. Given a simplicial complex, we present an algorithm which reduces the number of its vertices, keeping its homology (i.e. connectivity, coverage) unchanged. We show that the algorithm reaches a Nash equilibrium, moreover we find both a lower and an upper bounds for the number of vertices removed, the complexity of the algorithm, and the maximal order of the resulting complex for the coverage problem. We also give some simulation results for classical cases, especially coverage complexes simulating wireless sensor networks.
Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.
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