2026 Seminars

Abstract: We consider expectations of the form $E [tr h_1(X_1^N)... tr h_r(X_r^N)]$, where $X_i^N$ are self-adjoint polynomials in various independent classical random matrices and $h_i$ are smooth test function and obtain a large $N$ expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials. This talk is based on joint work with Benoît Collins.

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: In this talk, I will present the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. A key focus of the talk will be the surgery method we developed to handle these singularities and establish global asymptotics. I will also discuss applications of this result, including the convergence of the characteristic polynomial of random normal matrices to Gaussian Multiplicative Chaos measure. Based on joint work with Paul Bourgade, Guillaume Dubach, and Lisa Hartung.

Abstract: TBA

Abstract: TBA

Abstract: Central limit theorems for both the logarithm of unitary characteristic polynomials, and for the logarithm of the Riemann zeta function are classical in their respective fields.  Recently, more attention has been directed to understanding atypical values by studying large deviations of these random variables.  One avenue of our recent work has been to study the transition between the typical-atypical regimes for the random matrix polynomials, as well as their derivatives. Another has been to prove conditional upper bounds on deviation probabilities for the Riemann zeta function, which as a consequence recovers the best-known bounds for its moments.  Both are achieved by taking a probabilistic perspective, and drawing connections with random walks. This talk includes joint work with L.-P. Arguin and A. Roberts, and S. Ortiz.

Abstract: I will present recent results on the winding number of determinantal curves defined as the determinant of an additive two-matrix field evaluated along the unit circle. I will discuss finer spectral features, such as the cycle structure of the eigenvalue flow as the base parameter winds around the unit circle as well as the distribution of exceptional points.  An exact formula for the associated partition function will be presented, followed by a description of the asymptotic behaviour of the winding number, when the source matrices are drawn from a subclass of bi-unitarily invariant ensembles.

This is based on joint work with Mario Kieburg.

Abstract: We prove an optimal global rigidity estimate for the eigenvalues of the Jacobi unitary ensemble. Our approach begins by constructing a random measure defined through the eigenvalue counting function. We then prove its convergence to a Gaussian multiplicative chaos measure, which leads to the desired rigidity result. To establish this convergence, we apply a sufficient condition from Claeys et al.  (Duke Math. J. 2021) and conduct an asymptotic analysis of the related exponential moments. This is a joint work with Chenhao Lu. 

Abstract: This talk focuses on the spectral properties of Hermitian K×K-block random matrices with independent centred entries and block-dependent variances. Such matrices are useful for modelling inhomogeneous systems, e.g. clustered random graphs with different coupling strengths within and between K clusters. It is known that when the variance profile is sparse (i.e. the random matrix contains many zero blocks), the spectral density develops a singularity at the origin as the dimension of the blocks goes to infinity.

 We compute the microscopic scaling limit of the density at the origin. For a low number of blocks (K=2 and K=3) we find that it depends only on the pattern of zero blocks but not on the specific values of the variances. A complete classification of possible limits for arbitrary K is in progress. Our derivation is based on an exact integral expression for the Stieltjes-transform of the density that we obtain using anti-commuting variables and the superbosonization formula. The scaling limit then follows via a saddle-point approximation.

Based on joint work with Torben Krüger (arXiv:2511.19308).

Abstract: I will discuss some new results concerning local scaling limits for eigenvalues of random normal matrices. In particular we obtain new universality results for the limiting rescaled kernel at hard edges without symmetry assumptions on the potential or the hard edge. We also obtain universality for soft/hard edges and extend known results for soft edges to disconnected droplets. The results are based on a direct approach, which avoids the use of orthogonal polynomials. The main ingredients are instead Paley-Wiener theorems for Hilbert spaces of entire functions associated with the limiting kernel and a construction of weighted polynomials with properties mimicking the properties of the correlation kernel.

Abstract: In this talk we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of dimension $n$ and order-$r$, drawn from the density \[ f(\mathcal{G}) = \frac{1}{Z_r(n)} \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg).\] 

 We consider contractions of the form $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain a Baik-Ben Arous-P\'{e}ch\'{e} phase transition for the largest and the smallest eigenvalues of such contractions at $r = 3$. We also show that the extreme eigenvectors contain non-trivial information about $\mathbf w$. In fact, in the regime $1 \ll r \ll n$, there are two vectors, one of which is perfectly aligned with $w$. We also obtain some results on mixed contractions $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ in the case $r = 4$. While the total variation distance between the joint distribution of the entries of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ and that of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{u}$ goes to $0$ when $\|\mathbf{u} - \mathbf{v}\| = o(n^{-1})$, the bulk and the largest eigenvalues of these matrices have the same limit profile as long as $\|\mathbf{u} - \mathbf{v}\| = o(1)$. Further, it turns out that there are no outlier eigenvalues when $\langle \mathbf{u}, \mathbf{v}\rangle = o(1)$. This talk is based on a joint work with Soumendu Sundar Mukherjee.

Abstract: For a square matrix, the range of its Rayleigh quotients is known as the numerical range, which is a compact and convex set by the Toeplitz–Hausdorff theorem. The largest value in this convex set is known as the numerical radius, which is often used to study the convergence rate of iterative methods for solving linear systems. In this talk, we will introduce a recent result on the asymptotic behavior of the numerical radius of a large-dimensional, complex, non-Hermitian random matrix and its elliptic variants. For the former, remarkably, the radius can be represented as the extremum of a stationary Airy-like process, which undergoes a correlation-decorrelation transition from a small to a large time scale. Based on this transition, we obtain the precise first and second order terms of the numerical radius. In the elliptic case, we show that the fluctuation of the numerical radius reduces to the maximum or minimum of two independent Tracy-Widom variables. Based on joint work with Giorgio Cipolloni.