Seminar on Free Probability
and Random Matrices

Winter 2018

Organizer: J. Mingo


Tuesday, February 13, 3:30 - 5:00, Jeff 319
Jamie Mingo (Queen's)
The Infinitesimal Law of the GOE, Part II
If $X_N$ is the $N \times N$ Gaussian Orthogonal Enemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

Tuesday, February 6, 3:30 - 5:00, Jeff 319
Jamie Mingo (Queen's)
The Infinitesimal Law of the GOE
If $X_N$ is the $N \times N$ Gaussian Orthogonal Enemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.


Tuesday, January 30, 3:30 - 5:00, Jeff 319
Neha Prabu (Queen's)
Semicircle distribution in number theory, Part II
In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

Tuesday, January 23, 3:30 - 5:00, Jeff 319
Rob Martin (University of Cape Town)
A multi-variable de Branges-Rovnyak model for row contractions
In the operator-model theory of de Branges and Rovnyak, any completely non-coisometric (CNC) contraction on Hilbert space is represented as the adjoint of the restriction of the backward shift to a de Branges-Rovnyak subspace of the classical (vector-valued) Hardy space of analytic functions in the open unit disk.

We provide a natural extension of this model to the setting of CNC (row) contractions from several copies of a Hilbert space into itself. A canonical extension of Hardy space to several complex dimensions is the Drury-Arveson space, and the appropriate analogue of the adjoint of the restriction of the backward shift to a de Branges-Rovnyak space is a Gleason solution, a row contraction whose adjoint acts as a several-variable difference quotient. Our several-variable model completely characterizes the class of all CNC row contractions which can be represented as (extremal contractive) Gleason solutions for a multi-variable de Branges-Rovnyak subspace of (vector-valued) Drury-Arveson space.


Tuesday, January 16, 4:00 - 5:30, Jeff 319
Neha Phabu (Queen's University)
Semicircle distribution in number theory
In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

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Getting to Jeffery Hall from the Hotel Belvedere