Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Free Probability and Random Matrices Seminar

Probability Seminar - Camille Male (Bordeaux)

Tuesday, April 10th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Camille Male (Bordeaux)

Title: An introduction to traffic independence

Abstract: 

The properties of the limiting non-commutative distribution of random matrices can be usually understood thanks to the symmetry of the model, e.g. Voiculescu's asymptotic free independence occurs for random matrices invariant in law by conjugation by unitary matrices. The study of random matrices invariant in law by conjugation by permutation matrices requires an extension of free probability, which motivated the speaker to introduce in 2011 the theory of traffics. A traffic is a non-commutative random variable in a space with more structure than a general non-commutative probability space, so that the notion of traffic distribution is richer than the notion of non-commutative distribution. It comes with a notion of independence which is able to encode the different notions of non-commutative independence.

The purpose of this task is to present the motivation of this theory and to play with the notion of traffic independence.

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Probability Seminar - Pei-Lun Tseng (Queen's University)

Tuesday, April 3rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Pei-Lun Tseng (Queen's University)

Title: Linearization trick of infinitesimal freeness II

Abstract:  Last week, we introduced how to find a linearization for a given selfadjoint polynomial and showed some properties of this linearization. We will continue our discussion this week and introduce the operator-valued Cauchy transform. Then, we will show the algorithm for finding the distribution of $P$ where $P$ is a selfadjoint polynomial with selfadjoint variables $X$ and $Y$. Based on this method, we will discuss how to extend this algorithm for finding infinitesimal distribution for $P$.

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Probability Seminar - Pei-Lun Tseng (Queen's University)

Tuesday, March 27th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Pei-Lun Tseng (Queen's University)

Title: Linearization trick of infinitesimal freeness

Abstract:  For given infinitesimal distribution of selfadjoint elements $X$, $Y$, and given a selfadjoint polynomial $P$ with variable $X$ and $Y$. The natural question is whether we can write down the precise formula for the infinitesimal distribution of $P$? In 2009 Belinschi and Shlyakhtenko gave a precise formula to solve for the infinitesimal distribution of $P$ for $P(X,Y)=X+Y$. In the talk, we will discuss how to find the formula for an arbitrary polynomial by using the linearization trick.

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Probability Seminar - Mihai Popa (University of Texas, San Antonio)

Tuesday, March 20th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Mihai Popa (University of Texas, San Antonio)

Title: Permutations of Entries and Asymptotic Free Independence for Gaussian Random Matrices

Abstract:  Since the 1980's, various classes of random matrices with independent entries were used to approximate free independent random variables. But asymptotic freeness of random matrices can occur without independence of entries: in 2012, in a joint work with James Mingo, we showed the (then) surprising result that unitarily invariant random matrices are asymptotically (second order) free from their transpose. And, in a more recent work, we showed that Wishart random matrices are asymptotically free from some of their partial transposes. The lecture will present a development concerning Gaussian random matrices. More precisely, it will describe a rather large class of permutations of entries that induces asymptotic freeness, suggesting that the results mentioned above are particular cases of a more general theory.

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Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, March 6th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of a real Wishart Matrix

Abstract:  The Wishart ensemble is the random matrix ensemble used to estimate the covariance matrix of a random vector. Infinitesimal freeness is a generalized independence stronger than freeness but weaker than second order freeness. I will give the infinitesimal distribution of a real Wishart matrix. It is given in terms of planar diagrams which are 'half' of a non-crossing annular partition.

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Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 13th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of the GOE, Part II

Abstract:  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.

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Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 6th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of the GOE

Abstract:  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.

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Free Probability Seminar - Neha Prabu (Queen's University)

Tuesday, January 30th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Neha Prabu (Queen's University)

Title: Semicircle distribution in number theory, Part II

Abstract:  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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Free Probability Seminar - Rob Martin (University of Cape Town)

Tuesday, January 23rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Rob Martin (University of Cape Town)

Title: Non-commutative Clark measures for Free and Abelian multi-variable Hardy space

Abstract:  In classical Hardy space theory, there is a natural bijection between the Schur class of contractive analytic functions in the complex unit disk and Aleksandrov- Clark measures on the unit circle. A canonical several-variable analogue of Hardy space is the Drury-Arveson space of analytic functions in the unit ball of d-dimensional complex space, and the canonical non-commuting or free multi- variable analogue of Hardy space is the full Fock space over d-dimensional complex space. Here, the full Fock space is naturally identified with a non- commutative reproducing kernel Hilbert space of free or non-commutative ana- lytic functions acting on a several-variable non-commutative open unit ball. We will extend the concept of Aleksandrov-Clark measure, the bijection between the Schur class and AC measures, Clark’s unitary perturbations of the shift, Lebesgue decomposition formulas and additional related results from one to several commuting and non-commuting variables.

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