Martina, Silvia (2017) Spectra of random matrices. [Magistrali biennali]
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The recent interest of the scientific community about the properties of networks is based on the possibility to study complex real world systems by renouncing the exact knowledge of the nature of system itself. This approach allows to model the system, for example, as a large collection of agents linked together in pairs to form a network. The networks are very studied in different scientific fields and, particularly, in ecological one, in order to understand the dynamics of the evolution related to a community composed by different species interacting with each other. A random matrix can incorporate many information according to the type of the system. By using the graph’s theory, it is possible to extrapolate information about the matrix and, therefore, about the system considered. The statistical features of the eigenvalues of large random matrices have been the focus of wide interest in mathematics and physics. This thesis is mainly focused on the study of the spectral density of sparse random matrices. Symmetric random matrices and non-Hermitian matrices have been considered in this work, paying attention to both the analytical and numerical approach of the eigenvalues distribution calculation. There are different mathematical methods used to analyze ensembles of random matrices with a particular underlying symmetry. It is well-known that the spectral density of random matrices ensembles will converge, as the matrix dimensions grows, to a precise limit. One example is Girko elliptic law. The introduction of the sparsity is one of the factors that complicate enormously the mathematical analysis and new techniques for the calculation of the spectral density are welcome. The cavity method is a new approach presented to extend our knowledge about large-scale statistical behavior of eigenvalues of random sparse Hermitian and non-Hermitian matrices. Therefore, the cavity method provides a specific analysis related to the study about how the modularity structure influences the stability in the ecological communities.
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