2026 Seminars

Abstract: TBA

Abstract: In 1978, French, Mello and Pandey suggested a phenomenological "1/6" relation between number variance and the variance of the $L$-th ordered eigenvalue. While the physical origin of this relation is well understood in terms of spectral rigidity, the universal $1/6$ difference between the two variances remained a point of controversy. In this talk, we shall revisit this long-standing heuristic observation and show that it is, in fact, asymptotically exact for the $\beta=2$ Dyson symmetry class.

Beyond proving the universal $1/6$ limit, we shall determine the explicit convergence trajectory as $L \to \infty$. The key to our proof is a previously unknown sum rule for level-spacing autocovariances, whose derivation hinges on our previous work on the power-spectrum description of eigenvalue fluctuations. Analytical results for $\beta=2$ are complemented by their conjectural extensions to the $\beta=1$ and $\beta=4$ symmetry classes and further corroborated by comprehensive numerical analysis.

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: The Random Normal Matrix model is a probability measure on the space of normal matrices. By considering their eigenvalues one obtains a random set of points on the complex plane, which tend to accumulate in a certain compact subset called the droplet. More formally, as the number of points increases the empirical measure converges weakly to the associated equilibrium measure. This talk will focus on quantifying the equidistribution using the expected $2$-Wasserstein distance, a metric over the space of finite measures defined using optimal transport. The main tool is to regularize the empirical measure using the heat equation with Neumann boundary conditions on a suitable subset of the droplet.

Abstract: In this talk, we consider the local statistics of orthogonal polynomial ensembles near a hard edge, subject to a multiplicative deformation of the measure. Probabilistically, this deformation corresponds to a position-dependent conditional thinning of the particles. We prove that, under critical hard edge scaling and for a large class of potentials and deformation symbols, the correlation kernel of the conditional ensemble converges to a universal limit, which we identify as the conditional thinned Bessel point process. We derive an explicit expression for this limiting kernel in terms of the solution to a nonlocal integrable system depending on a parameter. For a special choice of the parameter, this system was recently identified in the study of multiplicative statistics of the Bessel point process. Our results establish that this system governs the full correlation structure of the conditional Bessel point process, extending the classical connection between the standard Bessel kernel and the Painlevé V equation. 

 This talk is based on a joint work with Leslie Molag and Guilherme L. F. Silva.

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: I discuss the widely used plasma analogy, which relates two seemingly unrelated problems. The first is the description of the quantum Hall effect for a two-dimensional electron gas in a strong magnetic field. The second is that of classical Coulomb gases with logarithmic interactions studied in particular in the context of random matrices. I will explain the many exact similarities between the two, but also how their edge properties might subtly differ.

Based mostly on 2407.19013

Abstract: The complexity of a system e.g one with many body interactions, in general, makes it difficult to determine some or almost all matrix elements (even in absence of disorder). The lack of accuracy acts as a source of randomness for the matrix elements and it can be modelled by a random matrix (with some or all random entries). While the statistical behaviour of a system in ergodic regime can be well-modelled by the stationary, basis-invariant random matrix ensembles, the non-ergodic regime requires consideration of basis-dependent ones: those which can take into account the physical constraints on the system e.g. local interactions, dimensionality, symmetry, local conservation laws. The statistical analysis of complex systems requires, therefore, a thorough probing of a wide range of system-dependent random matrix ensembles which is not an easy task. The theoretical analysis is further complicated by the fluctuation of accuracy due to varying system conditions causing multi parametric evolution of the matrix elements. Fortunately our theoretical analysis reveals that, for single-well potentials, it is possible to identify a function of all distribution parameters, seemingly a measure of average uncertainty and therefore referred as the complexity parameter, in terms of which the multi-parametric evolution can be reduced to a single parametric Dyson’s Brownian dynamics. As a consequence, the evolution of the statistical properties of a wide range of system-dependent random matrix ensembles can be described by a common mathematical formulation where system information enters through the complexity parameter only. This in turn suggests a possible classification of complex systems in an infinite range of universality classes characterised just by the complexity parameter and the nature of global physical constraints e.g. symmetries and conservation laws. The complexity parametric formulation of the statistical properties not only helps in technical simplification (one can apply results available for one complex system to another if they have same complexity parameter and global constraints), it also indicates a web of connection hidden underneath complex systems even in non-ergodic regime, thereby generalising the previously known universality in complex systems in ergodic regime).

Abstract: In this talk, I will present a $q$-deformed random unitary ensemble associated with the little-$q$ Laguerre weight, which provides a discrete analogue of the Laguerre unitary ensemble. In the double scaling regime $q=e^{-\lambda/N}$ where $N$ is the system size and $\lambda$ is an additional parameter, the limiting spectral distribution yields the $q$-deformation of the Marchenko-Pastur law. The key feature of the limiting density is the presence of a phase transition depending on the value of $\lambda$, due to an upper constraint. I will discuss three complementary approaches to the limiting density: the method of moments, the analysis of a constrained equilibrium problem, and the asymptotic zero distribution of $q$-orthogonal polynomials. In addition, I will introduce the underlying combinatorial structure of the spectral moments.

This talk is based on joint work with Sung-Soo Byun and Guido Mazzuca.

Abstract: Stein's method is a powerful technique for bounding the distance between two probability distributions with respect to a probability metric. Stein's method has proven to be a powerful tool for deriving quantitative probabilistic limit theorems, and has recently found exciting statistical applications beyond the traditional application to distributional approximation. In this talk we begin by reviewing some basic theory on Stein's method for normal approximation. We then see how this theory can be extended to develop Stein's method for matrix normal approximation, which represents the first systematic treatment of Stein's method for matrix-variate distributions. To illustrate this framework, we provide three applications, these being smooth Wasserstein distance bounds to quantify the matrix central limit theorem, a Wasserstein distance bound for the matrix normal approximation of the matrix T distribution, and the derivation of Stein’s method-of-moments estimators for scale parameters of the matrix normal distribution. We close by considering some further directions for research on Stein's method for matrix-variate distributions. This is joint work with Frederic Ouimet and Donald Richards. 

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: In this talk, we explore the global fluctuations of Gaussian elliptic matrices through both combinatorial and analytic methods. First, we introduce a spoke-arc decomposition of non-crossing annular pair partitions that records spoke type and orientation, isolates spoke-level contributions, and factorizes the dependence on the ellipticity parameter into a spoke factor and arc weights. This approach allows us to prove that an independent family of Gaussian elliptic matrices is asymptotically second-order free. We then complement these combinatorial results with an analytic framework, expressing the limiting covariance of linear statistics of the Gaussian elliptic matrix $X_N$ and those of $X_N X_N^*$ using explicit contour integrals and real domain integrals.

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.