Free Probability as a Possible Bridge between Random Matrices and Hecke Operators
自由槪率論作為隨機矩陣與Hecke 算子之間可能的橋樑
Student thesis: Master's Thesis
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Award date  9 Nov 2020 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(eb321d42281c499cba5904717aa17553).html 

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Abstract
Inspired by the similarity of Wigner's Semicircle Law in Random Matrix Theory and the semicircle distribution of Free Central Limit Theorem (FCLT) in Free Probability Theory, D. Voiculescu revealed that the connection hinges on two facts: first, a random matrix is generally asymptotically a semicircular element in a free probability space, secondly, several random matrices are generally asymptotically free. Here starting from the similarity of SatoTate distribution in Modular Forms Theory and the semicircle distribution of FCLT in Free Probability Theory, we showed that analogous results hold: first, a matrix which essentially contains all the Hecke operators on the modular space S_{k }(Γ_{0}(N ) of a fixed level N is an operatorvalued semicircular element in an operatorvalued free probability space, secondly, several matrices arising from Hecke operators on S_{k} (Γ_{0}(N ) of different level N are asymptotically operatorvalued free. Combining the work of D.Voiculescu and the results here, we may connect random matrices and Hecke operators via free probability.