The Jacobi Ensembles of random matrices have joint distribution of eigenvalues proportional to the integration measure in the Selberg integral. They can also be realised as the singular values of principal submatrices of random unitaries. In this talk we will review some old and new results concerning the distribution of the largest and smallest eigenvalues.

In 2004, Diaconis and Gamburd computed statistics of secular coefficients in the circular unitary ensemble. They expressed the moments of the secular coefficients in terms of counts of magic squares. Their proof relied on the RSK correspondence. We'll present a combinatorial proof of their result, involving the characteristic map. The combinatorial proof is quite flexible and can handle other statistics as well. We'll connect the result and its proof to old and new questions in number theory, by formulating integer and function field analogues of the result, inspired by the Random Matrix Theory model for L-functions.

Partly based on the arXiv preprint https://arxiv.org/abs/2102.11966

We consider, in a framework of iterated random conformal maps, a two-parameteraggregation model of Hastings-Levitov type, in which the size and intensity of new particles are each chosen to vary as a power of the density of harmonic measure. Then we consider a limit
in which the overall intensity of particles become large, while the particles themselves become
small. For a certain `sub-critical' range of parameter values, we can show a law of large numbers and fluctuation central limit theorem. The admissible range of parameters includes an off-lattice version of the Eden model, for which we can show that disk-shaped clusters are stable.
Many open problem remain, not least because the limit PDE does not yet have a satisfactory mathematical theory.

This is joint work with Vittoria Silvestri and Amanda Turner.

In this talk we will discuss the correlations of the Riemann Zeta in various ranges, and prove a new result for correlations of squares. This problem is closely related to correlations of the characteristic polynomial of CUE with a very subtle difference. We will explain where this difference comes from, and what it means for the moments of moments of the Riemann Zeta, and its maximum in short intervals.

Characterising the statistical properties of high dimensional random functions has been one of the central focus of the theory of disordered systems, and notably spin glasses, over the last decades. Applications to machine learning via deep neural network has seen a resurgence of interest towards this problem in recent years. The simplest yet non-trivial quantity to characterise these landscapes is the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points (or equilibria) with the dimension of the underlying space. A paradigmatic model for such random landscape in the $N$-dimensional Euclidean space consists of an isotropic harmonic confinement and a Gaussian random function, with rotationally and translationally invariant covariance [1]. The total annealed complexity in this model has been shown to display a ”topology trivialisation transition”: for weak confinement, the number of stationary points is exponentially large (positive complexity) while for strong confinement there is typically a single stationary point (zero complexity).

In this talk, I will present recent results obtained for a distinct exactly solvable model of random lanscape in the $N$-dimensional Euclidean space where the random Gaussian function is replaced by a superposition of $M > N$ random plane waves [2]. In this model, we compute the total annealed complexity in the limit $N\rightarrow\infty$ with $\alpha = M/N$ fixed and find, in contrast to the scenario exposed above, that the complexity remains strictly positive for any finite value of the confinement strength. Hence, there is no ”topology trivialisation transition” for this model, which seems to be a representative of a distinct class of universality.

References:

[1] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[2] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, arXiv preprint arXiv:2202.03815, submitted to J. Math. Phys.

The sinoatrial node (SAN) is the pacemaker region of the heart.
Recently calcium signals, believed to be crucially important in heart
rhythm generation, have been imaged in intact SAN and shown to be
heterogeneous in various regions of the SAN. However, calcium imaging
is noisy, and the calcium signal heterogeneity has not been
mathematically analyzed to distinguish meaningful signals from
randomness or to identify signalling regions in an objective way. In
this work we apply methods of random matrix theory (RMT) developed for
financial data and used for analysis of various biological data sets
including β-cell collectives and EEG data. We find eigenvalues of the
correlation matrix that deviate from RMT predictions, and thus are not
explained by randomness but carry additional meaning. We use
localization properties of the eigenvectors corresponding to high
eigenvalues to locate particular signalling modules. We find that the
top eigenvector captures a common response of the SAN to action
potential. In some cases, the eigenvector corresponding to the second
highest eigenvalue appears to yield a possible pacemaker region as its
calcium signals predate the action potential. Next we study the
relationship between covariance coefficients and distance and find
that there are long range correlations, indicating intercellular
interactions in most cases. Lastly, we perform an analysis of nearest
neighbor eigenvalue distances and find that it coincides with the
universal Wigner surmise. On the other hand, the number variance,
which captures eigenvalue correlations, is a parameter that is
sensitive to experimental conditions. Thus RMT application to SAN
allows to remove noise and the global effects of the action potential
and thereby isolate the correlations in calcium signalling which are
local. This talk is based on joint work with Chloe Norris with a
preprint found here:
https://www.biorxiv.org/content/10.1101/2022.02.25.482007v1.

The characteristic polynomial of a random Hermitian matrix induces naturally a field on the real line. In the case of the Gaussian Unitary ensemble (GUE), this fields is expected to have a very special correlation structure: the logarithm of this field is log-correlated and its maximum is at the heart of a conjecture from Fyodorov and Simm predicting its asymptotic behavior.   As a first step in this direction, we obtained in collaboration with R. Butez and O. Zeitouni, a central limit theorem for the logarithm of the characteristic polynomial of the Gaussian beta Ensembles and for a certain class of random Jacobi matrices. In this talk, I will explain how the tridiagonal representation of the GUE and orthogonal polynomials techniques allow us to analyse the fluctuations of the characteristic polynomial.

The dynamics of quantum many-body systems is intricately related to random matrix theory (RMT), to such a degree that quantum chaos is even defined through random matrix level statistics. However, exact results on this connection are typically precluded by the exponentially large Hilbert space. After a short introduction to the role of RMT in many-body dynamics, I will show how dual-unitary circuits present a minimal model of quantum chaos where this connection can be made rigorous. This will be illustrated using a new kind of emergent random matrix behaviour following a quantum quench: starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design, which can be shown to hold exactly in dual-unitary circuit dynamics.

Since their introduction in 1928 by Paul A. Dirac, Dirac operators have been playing essential roles in many areas of Physics and Mathematics. In particular, they provide powerful and efficient tools to clarify (and sometimes solve) important problems in representation theory of real Lie groups, p-adic groups or Hecke algebras, such as classification, unitarity and geometric realization. In this representation theoretic context, the Dirac index of a Harish-Chandra module is a virtual module induced by Vogan’s Dirac cohomology. Once we observe that Dirac index commutes with translation functors, we will associate a polynomial (on a Cartan subalgebra) with a coherent family of Harish-Chandra modules. Then we shall explain how this polynomial can be used to connect nilpotent orbits, associated cycles and the leading term of the Taylor expansion of the characters of Harish-Chandra modules. This is joint wok with P. Pandzic, D. Vogan and R. Zierau.