The main question that we will investigate in this talk is: what does the spectrum of a quantum channel typically looks like? We will see that a wide class of random quantum channels generically exhibit a large spectral gap between their first and second largest eigenvalues. This is in close analogy with what is observed classically, i.e. for the spectral gap of transition matrices associated to random graphs. In both the classical and quantum settings, results of this kind are interesting because they provide examples of so-called expanders, i.e. dynamics that are converging fast to equilibrium despite their low connectivity. We will also present implications in terms of typical decay of correlations in 1D many-body quantum systems. If time allows, we will say a few words about ongoing investigations of the full spectral distribution of random quantum channels. This talk will be based on: arXiv:1906.11682 (with D. Perez-Garcia), arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368 (with P. Oliveira Santos and P. Youssef).[-]

Haagerup and Thorbjørnsen proved that iid GUEs converge strongly to free semicircular elements as the dimension grows to infinity. Motivated by considerations from quantum physics -- in particular, understanding nearest neighbor interactions in quantum spin systems -- we consider iid GUE acting on multipartite state spaces, with a mixing component on two sites and identity on the remaining sites. We show that under proper assumptions on the dimension of the sites, strong asymptotic freeness still holds. Our proof relies on an interpolation technology recently introduced by Bandeidra, Boedihardjo and van Handel. This is a joint work with Benoît Collins.[-]

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom's original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.[-]

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom's original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.[-]

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom's original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.[-]

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom's original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.[-]

For the commonly studied Hermitian random matrix models there exist tridiagonal matrix models with the same eigenvalue distribution and the same spectral measure $v_{n}$ at the vector $e_{1}$. These tridiagonal matrices give recurrence coefficients that can be used to build the family of random polynomials that are orthogonal with respect to νn. A similar bijection between spectral data and recurrence coefficients also holds for the Unitary ensembles. This time in stead of obtaining a tridiagonal matrix you obtain a sequence $\left \{ \alpha _{k} \right \}_{k=0}^{n-1}$ Szegö coefficients. The random orthogonal polynomials that are generated by this process may then be used to study properties of the original eigenvalue process.
These techniques may be used not just in the classical cases, but also in the more general case of $\beta $-ensembles. I will discuss various ways that orthogonal polynomials techniques may be applied including to show convergence of the Circular $\beta $-ensemble to $Sine_{\beta }$. I will finish by discussing a result on the maximum deviation of the counting function of Sineβ from it expected value. This is related to studying the phases of associated random orthogonal polynomials.[-]

I will talk about a transformation involving double monotone Hurwitz numbers, which has several interpretations: transformation from maps to fully simple maps, passing from cumulants to free cumulants in free probability, action of an operator in the Fock space, symplectic exchange in topological recursion. In combination with recent work of Bychkov, Dunin-Barkowski, Kazarian and Shadrin, we deduce functional relations relating the generating series of higher order cumulants and free cumulants. This solves a 15-year old problem posed by Collins, Mingo, Sniady and Speicher (the first order is Voiculescu R-transform). This leads us to a general theory of 'surfaced' freeness, which captures the all order asymptotic expansions in unitary invariant random matrix models, which can be described both from the combinatorial and the analytic perspective.
Based on https://arxiv.org/abs/2112.12184 with Séverin Charbonnier, Elba Garcia-Failde, Felix Leid and Sergey Shadrin.[-]

This talk will focus on the fluctuations of a linear spectral statistic around its mean for $P\left(W_N, D_N\right)$ where $P$ is a polynomial, $W_N$ a Wigner matrix and $D_N$ a deterministic diagonal matrix. I will first consider the case when $P\left(W_N,D_N\right)=W_N+D_N$, based on a joint work with M. Février (U. Paris-Saclay). In the general case of $P$ a selfadjoint noncommutative polynomial, I will present results for the special case of the Stieltjes transform, based on a joint work with S. Belinschi (CNRS, U. Toulouse), M. Capitaine (CNRS,U. Toulouse) and M. Février (U. Paris-Saclay).[-]