Probabilistic Operator Algebras Seminar

(University of Oxford)

Based on joint work with Davide Macera and Joe Thomas.

The first non-zero eigenvalue, or spectral gap of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about the surface. We study the size of the spectral gap for random large genus hyperbolic surfaces sampled according to the Weil-Petersson probability measure. We show that there is a c > 0 such that a random surface of genus g has spectral gap at least 1/4 – O(g^{-c}) with high probability. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel, and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel, to the Laplacian on Weil-Petersson random hyperbolic surfaces.

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(University of Notre Dame)

Biane’s free multiplicative Brownian motion b_t is the large-N limit of the Brownian motion in the general linear group GL(N;\mathbb{C}) and can be viewed as the solution to a free stochastic differential equation driven by a circular Brownian motion. I will consider random walk approximations to b_t which are discrete approximations to the solution of the SDE. These approximations have the form of a product of steps, each of which is the identity plus a multiple of a circular element. We are able to compute the Brown measure of the model with a fixed number of steps using the linearization method. We are then able to let the number of steps tend to infinity and recover the previously computed Brown measure of b_t itself. A key step in the argument is a new freeness result for block elements. In general, matrices with freely independent entries are not freely independent in the ordinary sense but only in the “operator valued” sense . But we show that in some interesting examples, we do obtain freeness in the ordinary sense. We also show that for a fixed number of steps, the empirical eigenvalue distribution of the corresponding matrix model converges to the Brown measure of the free model.

This is joint work with Bruce Driver, Ching Wei Ho, Todd Kemp, Yuriy Nemish, Evangelos Nikitopolous, and Félix Parraud.

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(University of Virginia)

I will talk about joint work with Kunnawalkam Elayavalli, Robert, and Patchell. In it, we introduce and study a natural notion of selflessness for inclusions of C*-probability spaces, which in particular necessitates all intermediate C*-algebras to be selfless in the sense of Robert. We identify natural sources of selfless inclusions in the realms of \mathcal{Z}-stable and free product C*-algebras. As an application of this, we prove selflessness for a new family of C*-probability spaces outside the regime of free products and group C*-algebras. These include the reduced free unitary compact quantum groups.

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(University of Wrocław)

We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a (q,t)-Poisson type operator (random variable) on the (q,t)-Fock space. From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the (q,t)-Charlier polynomials. These generalize the q-Charlier polynomials of Saitoh Yoshida and reflect deeper combinatorial symmetries through the additional deformation parameter t .A central feature of this paper is the derivation of a combinatorial moment formula of the (q,t)-Poisson type operator and the (q,t)-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the (q,t)-deformation.

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(UCLA)

In this joint work with Ben Major, we show that for a tracial von Neumann algebra M, a variety of decomposition properties (such as having a Cartan subalgebra, a non-trivial tensor product decomposition, or property Gamma) imply the existence of an amenable subalgebra N (in M, or possibly in its ultrapower) and a generating set X for M, such that the algebra generated by the
semicircular perturbation of the set X actually contains M itself. Using properties of Voiculescu’s non-microstates free entropy and a recent result of Jekel and Pi, this gives a short proof of strong 1-boundedness (in the sense of Jung) for such algebras.

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(University of Copenhagen, Michigan State University)

In joint work with Yoonkyeong Lee and Jennifer Pi, we establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices \eta = (\eta_{i,j})_{i,j \in I}. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of \eta_{i,j}, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a B-valued semicircular family and generators of the base algebra B. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.

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(Imperial College London)

When attempting to construct QFTs that include Fermions using the methods of Stochastic Quantisation, one is naturally forced to consider noncommutative stochastic PDEs. I shall show how to formulate SPDEs driven by noncommutative noises in terms of algebra-valued singular PDEs. Furthermore, I will describe how one can renormalise singular products appearing in such equations, using the example of free probability and q-Gaussian noises. Finally, I will show how one can derive operator insertion estimates, which are crucial in solving singular SPDEs, such as the \Phi_2^4-equation, by changing the topology on the algebra of q-Gaussian. This talk is based on joint work with Ajay Chandra and Martin Hairer.

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