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http://cimat.repositorioinstitucional.mx/jspui/handle/1008/973| Non-Commutative Probability and Random Matrices with Discrete Spectrum | |
| ADRIAN DE JESUS CELESTINO RODRIGUEZ | |
| Acceso Abierto | |
| Atribución-NoComercial | |
| Probabilidad libre | |
| In this thesis, we present a variation of Free Probability called Cyclic Monotone Independence introduced by Collins, Hasebe and Sakuma (2016), along with its application to the study of random matrix models with discrete spectrum. This work is divided into two parts. In the first one, we present the abstract notion of cyclic monotone independence on a non-commutative probability space $(\mathcal{A},\tau)$ with an additional linear functional $\omega$, and we enunciate analogous definitions such as moments, convergence in distribution and asymptotic cyclic monotone independence for the linear functional $\omega$. Then, we focus on the case that cyclic monotone elements have the same distribution of trace class operators, and we study explicit formulas for the eigenvalues of some selfadjoint polynomials on such cyclic monotone elements. We give an alternative proof by considering the respective polynomials as the (1,1)-entry of products of some matrices. This idea allows to find new formulas for other polynomials not considered in the paper of the three authors. The second part of the thesis is dedicated to study the fact that random matrices that converge to trace class operators and rotationally invariant random matrices having limiting non-commutative distribution are asymptotically cyclically monotone independent on average and almost surely. This allows to prove that polynomials on the previous kind of matrices converge to deterministic compact operators. Combining the formulas from first part and the asymptotic cyclic monotone independence of random matrices, we obtain formulas for the limiting multiset of eigenvalues of the selfadjoint polynomials considered before. | |
| 19-10-2018 | |
| Tesis de maestría | |
| PROBABILIDAD | |
| Versión aceptada | |
| acceptedVersion - Versión aceptada | |
| Aparece en las colecciones: | Tesis del CIMAT |
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| TE698.pdf | 929.84 kB | Adobe PDF | Visualizar/Abrir |