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).[-]

One of the most important question in Quantum Information Theory was to figure out whether the so-called Minimum Output Entropy (MOE) was additive. In this talk I will start by defining the counter-example originally built by Belinschi, Collins and Nechita. Then I will explain how with the help of a novel strategy, we managed with Collins to compute concentration estimate on the probability that the MOE is non-additive and how it yielded some explicit bounds for the dimension of spaces where violation of the MOE occurs. Finally, I will talk more in detail about this novel strategy which consists in interpolating random matrices and free operators with the help of free stochastic calculus.[-]

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.[-]

We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable solution procedure, as well as computational simulations to see dynamics of correlation functions in action.[-]

We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable solution procedure, as well as computational simulations to see dynamics of correlation functions in action.[-]

We study the expectation of the matrix of overlaps of left and right eigenvectors in the complex Ginibre ensemble, conditioned on a fixed number of k complex eigenvalues.
The diagonal (k=1) and off-diagonal overlap (k=2) were introduced by Chalker and Mehlig. They provided exact expressions for finite matrix size N, in terms of a large determinant of size proportional to N. In the large-N limit these overlaps were determined on the global scale and heuristic arguments for the local scaling at the origin were given. The topic has seen a rapid development in the recent past. Our contribution is to derive exact determinantal expressions of size k x k in terms of a kernel, valid for finite N and arbitrary k.
It can be expressed as an operator acting on the complex eigenvalue correlation functions and allows us to determine all local correlations in the bulk close to the origin, and at the spectral edge. The methods we use are bi-orthogonal polynomials in the complex plane and the analyticity of the diagonal overlap for general k.
This is joint work with Roger Tribe, Athanasios Tsareas, and Oleg Zaboronski as appeared in arXiv:1903.09016 [math-ph][-]

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.[-]

Usually one defines a Tau function Tau(t_1,t_2,...) as a function of a family of times having to obey some equations, like Miwa-Jimbo equations, or Hirota equations.
Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the space of cycles (cf Goldman bracket), so that Hamiltonians can be represented by cycles.
All the integrable system formalism can then be represented geometrically in the space of cycles: the Poisson bracket is the intersection, the conserved quantities are periods, Miwa-Jimbo equations and Seiberg-Witten equations are a mere consequence of the definition, Hirota equation is a vanishing monodromy condition, and Virasoro-W constraint are automatically satisfied by our definition, showing that our Tau-function is also a conformal block. Our definition contains KdV, KP multicomponent KP, Hitchin systems, and probably all known classical integrable systems.[-]

I will discuss polynomials $P_{N}$ of degree $N$ that satisfy non-Hermitian orthogonality conditions with respect to the weight $\frac{\left ( z+1 \right )^{N}\left ( z+a \right )^{N}}{z^{2N}}$ on a contour in the complex plane going around 0. These polynomials reduce to Jacobi polynomials in case a = 1 and then their zeros cluster along an open arc on the unit circle as the degree tends to infinity.
For general a, the polynomials are analyzed by a Riemann-Hilbert problem. It follows that the zeros exhibit an interesting transition for the value of a = 1/9, when the open arc closes to form a closed curve with a density that vanishes quadratically. The transition is described by a Painlevé II transcendent.
The polynomials arise in a lozenge tiling problem of a hexagon with a periodic weighting. The transition in the behavior of zeros corresponds to a tacnode in the tiling problem.
This is joint work in progress with Christophe Charlier, Maurice Duits and Jonatan Lenells and we use ideas that were developed in [2] for matrix valued orthogonal polynomials in connection with a domino tiling problem for the Aztec diamond.[-]