Free Probability as a Possible Bridge between Random Matrices and Hecke Operators

Abstract

Inspired by the similarity of Wigner's Semi-circle Law in Random Matrix Theory and the semi-circle 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 semi-circular element in a free probability space, secondly, several random matrices are generally asymptotically free. Here starting from the similarity of Sato-Tate distribution in Modular Forms Theory and the semi-circle 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 (Γ₀(N ) of a fixed level N is an operator-valued semi-circular element in an operator-valued free probability space, secondly, several matrices arising from Hecke operators on S_k (Γ₀(N ) of different level N are asymptotically operator-valued free. Combining the work of D.Voiculescu and the results here, we may connect random matrices and Hecke operators via free probability.

Date of Award	9 Nov 2020
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Dennis AMELUNXEN (Supervisor), Weifeng QIU (Supervisor) & Guo LUO (External Co-Supervisor)