2024 Seminars

Abstract: I will talk about the infinite particle limit of eigenvalue stochastic dynamics introduced by Rider and Valko. These are the canonical dynamics associated to the inverse Laguerre ensemble in the way Dyson Brownian motion is related to the Gaussian ensemble. For this model we can prove convergence, from all initial conditions, to a new infinite-dimensional Feller process, describe the limiting dynamics in terms of an infinite system of log-interacting SDE that is out-of-equilibrium and finally show convergence in the long-time limit to the equilibrium state given by the (inverse points of the) Bessel determinantal point process.

Abstract: In the talk, I will give a brief review of known results on the extreme gap problems (smallest and largest gaps of the eigenvalues) of classical random matrix ensembles. Then I will present our recent series of work on smallest gaps of several random point processes with Gaussian structures.

Abstract: I will give an overview of some aspects of the theory of random matrices over finite fields.  We will touch on how this theory differs from classical random matrix theory and also on how some classical tools can be adapted to this setting.  Finally, we will discuss some recent developments that is joint work with Hoi Nguyen and Sean Meehan.

Abstract: The Markov--Krein transform with parameter $c > 0$ is a nonlinear transform between measures which is related to Dirichlet processes, representations of the symmetric group, and the Markov moment problem, to name a few. It can be used to define the $c$-convolution of probability measures interpolating the classical convolution and the free convolution. In this talk, we use the Markov--Krein transform to identify the limit in a high temperature regime of classical beta ensembles on the real line and related eigenvalue processes. For the static result, the limiting measure of Gaussian beta ensembles (resp. beta Laguerre ensembles and beta Jacobi ensembles) is the inverse Markov--Krein transform of the Gaussian distribution (resp. the gamma distribution and the beta distribution). At the process level, we show that the limiting probability measure-valued process is the inverse Markov--Krein transform of a certain 1d stochastic process. It is based on joint works with F. Nakano and H.D. Trinh.

Abstract: The ground state of a free Fermi gas is a classical example of determinantal processes whose correlation kernel is associated with a Schrödinger operator on R^n. This observation is due to Macchi (1975) and determinantal processes have been intensively studied since then. In this talk, I will explain recent results that we obtained with Alix Deleporte about the fluctuations of these determinantal point processes, this will include universality of local correlations and different central limit theorems. The proof are based on the semiclassical analysis of Schrödinger operator, and more generally pseudo-differential operators, on R^n.

Abstract: In this talk, I will firstly introduce some background on the problem of joint moments of the derivatives of  the characteristic polynomial of a random unitary matrix. Secondly, I will discuss the connections between joint moments in finite matrix size and the large matrix size limit, and solutions of the sigma-Painleve V and the sigma-Painleve III'  equations, respectively. Thirdly,  I will talk about the applications of these results, including explicit formulae for joint moments of finite matrix size, an efficient way to compute joint moments of power sum linear statistics of a certain determinantal process.

Abstract: We consider the normal matrix model with external potential N|z|^2 − 2c N log |z^2+ a^2|, which represents the Ginibre ensemble with two point charge insertions. As n, N → ∞ with n/N → t > 0,the eigenvalues fill out a bounded region in the complex plane, known as the droplet. The average characteristic polynomial satisfies planar orthogonality with respect to a weight supported over the entire complex plane. We show that, for a certain regime of parameters a, c, and t, the limiting zero-counting measure (motherbody) is supported on an interval along the real line, with an asymptotic density characterized by a vector equilibrium problem. We rely on a recent result by Berezin, Kuijlaars, and Parra, which allows us to reformulate the planar orthogonality in terms of non-Hermitian multiple orthogonality (Type I). This opens the door for the steepest descent analysis of the associated Riemann-Hilbert problem.

Abstract: In this talk, I will present a method for obtaining one-point lower tail large deviation principles (LDPs) for the stochastic six-vertex model. The proof combines connections between the stochastic six vertex model and discrete log-gases on the half line with novel methods to establish log-concavity estimates on tail probabilities. Based on two joint works and one ongoing collaboration with Sayan Das and Yuchen Liao.

Abstract: We expand on the relationship between random matrix and multiplicative chaos theories using the integrability properties of the circular beta-ensembles. We give a comprehensive proof of the multiplicative chaos convergence for the characteristic polynomial and eigenvalue counting function of the circular beta-ensembles throughout the subcritical phase, including negative powers. This generalizes recent results in the unitary case to any beta>0 and for the eigenvalue counting field.

Abstract: Coulomb gases consist of n particles repelling each other via the 2d Coulomb law and subject to the presence of an external potential. In this talk, I will discuss recent results on the probability that a given subset of the plane is free from particles when n is large. I will also talk about the most likely point configurations ("far from equilibrium") that produce such holes, which are described in terms of balayage measures.

Abstract: The dissipative spectral form factor (DSFF) has recently emerged as a diagnostic tool for assessing non-integrability or chaos in non-Hermitian systems. It extends the concept of the spectral form factor (SFF), a widely used measure for similar investigations in Hermitian systems. In this study, we concentrate on the elliptic Ginibre unitary ensemble (eGinUE), which interpolates between the non-Hermitian limit of Ginibre unitary ensemble (GinUE) and the Hermitian limit of Gaussian unitary ensemble (GUE) by varying a symmetry parameter. We derive exact finite-dimension expression and large-dimension approximation for the DSFF of eGinUE. A key finding of this work is a compelling “scaling relationship” between the DSFF of eGinUE and that of the GUE or GinUE. This scaling relationship also reveals a previously unobserved connection between the DSFF of GinUE and the SFF of GUE. The DSFF for eGinUE displays a dip-ramp-plateau structure in the GinUE and GUE limits, as well as in the crossover region, with differences in time scales that are well-explained by the aforementioned scaling relationship. Additionally, depending on the symmetry regime within the crossover, we provide various estimates for the Thouless time ($T_{Th}$) and Heisenberg time ($T_{H}$), which correspond to the dip-ramp and ramp-plateau transitions, respectively. Our analytical results are compared with Monte Carlo simulations of the eGinUE random matrix model, showing excellent agreement.

Abstract: The distribution P(r) of the ratio r of two consecutive level spacings has been employed as a measure to quantify chaoticity of quantum systems, alternative to the more conventional level spacing distribution that involves uncertainty due to unfolding. After reviewing our previous work on Janossy densities for the unitary ensembles, we present an analytic expression on P(r) for the sine kernel, in terms of the Tracy-Widom system of differential equations. As a showcase of the efficacy of our results for characterizing an approach to quantum chaoticity, we contrast them to the Riemann zeta zeroes on the critical line at increasing heights.

Abstract:  Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$.  We observe a phase transition for the extreme eigenvalues(in the sense of modulus) in deformed GinUE, both in eigenvalue statistics(via 1-point correlation function) and eigenvector statistics(via the mean self overlap function). 

 There are three regimes: (1) When all eigenvalues of $X_0$  lie in the open disk $D(0,\sqrt{\tau})$, local statistics are still governed by the GinUE statistics; (2) When some eigenvalues of $X_0$ are on the circle and all others lie inside it, both eigenvalue and eigenvector statistics can be characterized by the iterative erfc integrals; (3) When some eigenvalues of $X_0$ go outside the circle, outlier eigenvalues occur, and the eigenvalue and eigenvector statistics are characterized by new kinds of functions which are different from the edge statistics. 

 The contents are mainly based on "Phase transition of eigenvalues in deformed Ginibre ensembles. arXiv:2204.13171v2", joint work with Dang-Zheng Liu and "Mean eigenvector self-overlap in deformed complex Ginibre ensemble. arXiv:2407.09163v2".

Abstract: Shanks conjectured that the mean of the derivative of the Riemann zeta function evaluated at the zeros of zeta is real and positive. This was proven in the 1980s by Conrey and Ghosh by finding an asymptotic for the first discrete moment (which was real and positive). We will talk about various extensions of this to higher derivatives and higher moments, where random matrix insights have proven very useful. This work is joint with Andrew Pearce-Crump. The talk will be at a very relaxed level, and assumes no knowledge of number theory.

Abstract: Determinantal point processes (DPPs) are mathematical models of random interacting particles by repulsive forces, and they appear in various fields, such as statistical mechanics, random matrix theory, representation theory, and combinatorics. In particular, within the context of representation theory, DPPs on a discrete space are well studied. Moreover, it is known that operator algebras, called CAR algebras, and their special linear functionals, called gauge-invariant quasi-free states, give rise to discrete DPPs. In this talk, we will develop the relationship between DPPs and operator algebra theory and discuss how dynamics on operator algebras give rise to stochastic dynamics on DPPs.

Abstract: The calculation of the asymptotics of the probability that there are no particles in a certain gap, known as the gap probability, is an important problem in point processes. In this talk, I will present the asymptotic expansion of the gap probabilities of complex and symplectic induced spherical ensembles, which can be realized as determinantal and Pfaffian 2D Coulomb gases on the Riemann sphere with the insertion of point charges. More precisely, when the gap is a spherical cap around the poles, we compute the asymptotic behavior of gap probability up to O(1) term. Our proof relies on the uniform asymptotics of the incomplete beta function. This is based on joint work with Sung-Soo Byun.

Abstract: Free independence is at the heart of free probability, originating from a long-standing open classification problem of operator algebras. Free independence differs from the notion of independence between random variables, but it can be concretely realized by the asymptotic behavior of independent random matrices. There are many related results, and independence between random matrices is essential in obtaining free independence. A recent progress by Mingo and Popa in 2019 is the discovery of a fundamentally different approach to obtaining asymptotic freeness. They proved asymptotic freeness between the (four) partial transposes of a bipartite Wishart matrix without assuming independence between random matrices. In this talk, we explore why the partial transposition is considered important from the perspective of quantum information theory, and discuss the extendibility of this asymptotic freeness result to the general multipartite Wishart matrix and the associated free central limit theorem.

Abstract: The 2D Ising model is one of the most celebrated examples of an exactly solvable lattice model. Motivated by problems in statistical mechanics and 2D quantum gravity, in 1986 Vladimir Kazakov considered the Ising model on a random planar lattice using techniques from random matrix theory. He was able to derive a formula for the free energy of this model, and made the first prediction of the Kniznik-Polyakov-Zamolodchikov (KPZ) formula for the shift of the critical exponents of a conformal field theory when coupled to quantum gravity. Unfortunately, his derivation was not mathematically rigorous, and the formula he obtained for the free energy was somewhat unwieldy. In this talk, I will review some of the details regarding both the Ising model and random matrices, and sketch a rigorous proof of Kazakov’s formula for the free energy. If time permits, we will also discuss the multicritical behavior of this model. This is joint work with Maurice Duits and Seung-Yeop Lee.

Abstract: We will discuss new methods to, in principle, obtain all cumulants of von Neumann entropy over different models of random states. The new methods uncover the structures of cumulants in terms of lower-order joint cumulants involving families of ancillary linear statistics. Importantly, the new methods avoid the task of simplifying nested summations when using existing methods in the literature that becomes prohibitively tedious as the order of cumulant increases. This talk is based on an ongoing joint work with Youyi Huang.

Abstract: The matrix of eigenvector overlaps, introduced by Chalker & Mehlig, is known to have many applications, including the description of decay laws in quantum chaotic scattering and the characterization of eigenvalue sensitivity. For normal matrices, the corresponding eigenvector overlaps are trivial due to orthogonality. However, when one considers non-normal matrices, the entries of the matrix of overlaps can become macroscopically large. In this talk, we study the diagonal entries of the matrix of overlaps, denoted self overlaps, for ensembles of N ×N real and complex random matrices with varying degrees of non-normality. We focus in particular on the real and complex elliptic Ginibre ensembles, with mean zero i.i.d. Gaussian entries and a correlation between offdiagonal matrix entries, governed by τ ∈ [0, 1). We will present new results for the mean self-overlap associated with complex eigenvalues at finite N in both ensembles, however we are mainly concerned with large N asymptotic behaviour. As N becomes large, we consider three different regions of the complex plane with different density of complex eigenvalues: the spectral bulk, the spectral edge and a region of eigenvalue depletion close to the real line. This is done for two different limits of τ , known as strong non-Hermiticity, where τ ∈ [0, 1) is fixed as N → ∞ and weak non-Hermiticity, where τ → 1 as N → ∞. As part of this talk we will also review some important existing results in this field and provide numerical evidence of our new results.

Abstract: The eigenvalue probability density function of the Gaussian Unitary Ensemble permits a $q$-extension related to the discrete $q$-Hermite weight and the corresponding $q$-orthogonal polynomials. In this talk, I will review Flajolet and Viennot's classical theory concerning the combinatorics of the moments of orthogonal polynomials. This theory enabled us to derive a positive sum formula for the spectral moments of this model. This is based on joint work with Sung-Soo Byun and Peter Forrester.

Abstract: In the high-temperature regime for β-ensembles, the inverse temperature is set as β=O(1/N). In this setting, the entropic effects due to the integration Lebesgue measure play on the same scale as the energy. Due to a simultaneous energy minimization-entropy maximization, it results in the non-compactness of the support of the equilibrium measure. In previous research, in collaboration with Ronan Memin (IMT), we proved a CLT for linear statistics in this regime by inverting the master operator, which is a central object in the study of the fluctuations. In this presentation, I will demonstrate how to employ the loop equations analysis method to establish the existence of an asymptotic expansion for the log partition function. We will see how certain aspects of the proof are significantly more complex compared to the classical regime.

Abstract: The information paradox in quantum information exists since Hawking's proposal of black hole radiation. The fundamental question is: how is it possible to have a thermal radiation while the time evolution in a Hilbert space must be unitary? This physical question actually offers an interesting playground for random matrix theorists when studying the problem from a statistical mechanics' point of view. Already the very simple model by Page, who considered a uniformly distributed pure quantum state, gave rise for the fixed trace ensemble. Considering Hawking's description of black hole radiation via bosons one obtains a random matrix model involving the non-compact symplectic group. The relevant spectral quantities are not the eigenvalues but the symplectic eigenvalues of the corresponding covariance matrix. I will introduce the rather exotic concept of this kind of spectral statistics and report on the random matrix model we have studied.

Abstract: Two-dimensional Coulomb gases are studied when they are distributed on elliptic annuli and the asymptotic forms of the gas molecule correlation functions are evaluated. For that purpose, two-dimensional orthogonality of the Chebyshev polynomials on the complex plane is utilized.

Abstract: In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states, and describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our recent preprint, we provide a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, which corresponds to a lower bound on the geometric entanglement of random quantum states, and to a bound on the ground-state energy of a particular multispecies spherical spin glass model. Our result represents a first step towards solving an important question in quantum information that has been part of folklore.

Abstract: The talk is about the probabilistic relations between eigenvalues and singular values of bi-unitarily invariant ensembles. We first extend the notion of k-point correlation function to j,k-point correlation functions when studying the interactions between j singular values and k singular values and, then, give an exact formula for the 1,1-point correlation function. This formula simplifies drastically when assuming the singular values are drawn from a polynomial ensemble. We will give some idea of the proof for the main result. Finally, we will show some numerical simulations to illustrate what the 1,1-point correlation function looks like for the classical cases of Laguerre and Jacobi ensembles and what it reveals about the interactions between singular values and eigenvalues.

Abstract: In this talk, I will describe a particular class of biorthogonal measures related to discrete and semi-discrete polymers (Log-Gamma, O'Connell-Yor, and mixed). More precisely, I will show that the Laplace transform of the partition function of the mentioned polymers coincides with the multiplicative statistics of these biorthogonal measures. This result can be seen as a finite N variant of the connection between the narrow wedge solution of the KPZ equation and the Airy point process. It generalizes previous results of Imamura and Sasamoto for the (homogeneous) O'Connell-Yor polymer. Time permitting, I will show some applications to the small-temperature limit of these polymers and their relation with matrix models. These results have been obtained jointly with Tom Claeys.

Abstract: I will discuss recent work that determines the asymptotic distribution of the smallest gaps between complex eigenvalues of the real Ginibre ensemble. I will also provide a brief overview of what is currently known about extremal eigenvalue gaps of random matrices, both small and large, and highlight a few open problems. This talk is based on joint work with Matthew Meeker.

Abstract: The two-dimensional One-Component Plasma (OCP) is a Coulomb system that consists of identical, electrically charged particles embedded in a uniform background of the opposite charge, interacting through a logarithmic potential. In the 90s, Jancovici, Lebowitz and Manificat discovered a law for the probabilities of observing large charge fluctuations in the OCP. Mathematically, this law has only been fully proved in the determinantal case (i.e., for the Ginibre ensemble). A few years ago, Chatterjee introduced a hierarchical version of the OCP, inspired by Dyson's hierarchical model of the Ising ferromagnet. We show that the JLM law holds for the hierarchical Coulomb model at any finite positive temperature. 

Based on a joint work with Oren Yakir.

Abstract: For an $N\times N$ Hermitian matrix $A$  the eigenvalues of the top-left $N-1\times N-1$ submatrix (or truncation) of $A$ interlace with the original eigenvalues of $A$. We could then continue to remove rows and columns to get further interlacing sequences of eigenvalues, and we can think of this process as some kind of dynamics on the spectral measures. Similarly differentiating real rooted polynomials will produce interlacing among the roots, and we can think of this as some other dynamics on the root measures. Various recent results have shown, both heuristically and rigorously, that for random matrices these two processes produce the same dynamics on the measures. However, if we consider the analogous processes from non-Hermitian matrices or complex rooted polynomials there is no obvious geometric reason for the processes to coincide and the picture is much less clear. After looking at a brief history of the real case, we will discuss how one can connect these processes for single ring matrices and random polynomials with independent coefficients. This talk will be based on joint work with Sean O'Rourke and David Renfrew.

Abstract: We consider the random normal matrix model (the two-dimensional Coulomb gas at inverse temperature $\beta=2$). If the droplet is disconnected, there will be interactions between the different components leading to some new behaviours of the system. I will discuss some results in the case of a rotational-invariant potential where the droplet consists of several concentric annuli. In particular I will discuss how a disconnected droplet affects asymptotics of the partition function and the correlation kernel near the boundary.

The talk is based on work with Yacin Ameur and Christophe Charlier.

Abstract: Under the assumption of a finite fourth moment, the behavior of the smallest eigenvalue in covariance matrices has been extensively studied. In this talk, we explore the intriguing scenario where each entry possesses an infinite fourth moment while still having a finite second moment. We investigate the phase transition from Tracy- Widom to Gaussian fluctuation for the smallest eigenvalue in this context. Time-permitting, we will also delve into a more heavy-tailed regime, broadening the scope of our investigation. This is based on the joint work with Zhigang Bao and Xiaocong Xu.

Abstract: I will report on recent joint work with Kurt Johansson (KTH). We consider a Coulomb gas restricted to a Jordan domain in the complex plane. We ask how the asymptotic expansion of the free energy, as the number of particles tends to infinity, depends on the geometry of the domain. I will explain how this problem is related to the Grunsky operator -- a classical tool in complex analysis -- and how this in turn reveals a close connection to the Loewner energy and other interesting domain functionals. I will further discuss the effect of corners,  which turns out to be universal in a certain sense. Most main players (Grunsky, Loewner, etc) will be introduced in the talk.

Abstract: We establish, for every family of orthogonal polynomials in the Askey scheme and the q-Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall lattice. This generalizes to all families of orthogonal polynomials in the Askey scheme previous results of Corteel and Kim for the little q-Jacobi polynomials. This is joint work with Sylvie Corteel, Bhargavi Jonnadula, and Jon Keating.

Abstract: Muttalib-Borodin ensemble is defined by the joint probability density function $$\prod_{1 \leq i < j \leq n} (x_i - x_j)(x^{\theta}_i - x^{\theta}_j) \prod^n_{i = 1} e^{-nV(x_i)}.$$ It is proposed by physicist Muttalib as a toy model of quantum transport, and has relations to random matrix theory. Because of its simplicity and its non-trivial hard edge limit, the Muttalib-Borodin ensemble becomes the archetype of biorthogonal ensembles. Borodin studied this model in the $V(x) = x$ case, and found its limiting distribution around the hard edge $0$. We show that for a large class of $V$, the Muttalib-Borodin ensemble has the same limiting distribution, that is, the model has a universal property. Our approach is by the asymptomatic analysis of a kind of vector-valued Riemann-Hilbert problem, with a new construction of the local parametrix for irrational $\theta$.

Abstract: Considering a sum of a complex Ginibre matrix  and a deterministic matrix, we prove that there are  only two kinds of eigenvalue statistics  at the spectral edge: Ginibre edge  statistics and  critical statistics. The latter seems new.  The method depends on a key finding that the eigenvalue correlation functions can be expressed as matrix integrals  via  auto-correlations of  characteristic polynomials. This talk is based on joint work with Lu Zhang (USTC).

Abstract: We consider the ensemble of complex elliptic random matrices in the weakly non-Hermitian regime. First we prove that after adding a weakly non-Hermitian Gaussian matrix of comparatively small variance, the bulk correlation functions converge to a universal limit, under certain assumptions on the initial matrix. Using the method of time reversal we can remove the Gaussian component if the initial matrix has a sufficiently smooth distribution.

Abstract: The archetypal model of a non-Hermitian random matrix is the Ginibre ensemble, consisting of i.i.d. standard Gaussian entries with no symmetry constraints. Another interesting non-Hermitian ensemble is obtained by truncating a Haar distributed unitary matrix, which can be shown to recover the Ginibre ensemble in a suitable limit. I will discuss recent work which develops a character expansion approach for evaluating correlations of characteristic polynomials in such models. The approach leads to a number of dualities interchanging the size of the matrices with the number of factors in the correlator. At the same time, it gives an alternative route to finding closed form determinantal or Pfaffian expressions. This is joint work with A. Serebryakov (Sussex).

Abstract: Recently, overlap, which is defined by the left and right eigenvectors of a matrix, is one of the hottest topics in random matrix theory. This plays a role in measuring the non-Hermiticity of the matrix. Indeed, the overlap is trivial for Hermitian matrices, and hence, it plays an essential role for non-Hermitian matrices. In 2020, Akemann, Tribe, Tsareas, and Zaboronski showed that the k-th conditional expectation of the overlaps for the Ginibre unitary ensemble forms a determinantal structure. In this talk, based on their approach, I will show the determinantal structure of the overlaps for the induced Ginibre/spherical unitary ensembles. The former model is the generalization of the Ginibre unitary ensemble with the origin point insertion, and the latter model is the non-Gaussian model with the origin point insertion. I will also present the scaling limits for both models. The scaling limits in the strong non-unitary regime are the same as those shown by Akemann et al for the Ginibre unitary ensemble. As a consequence, the universality of the overlaps is confirmed. On the other hand, I will present new scaling limits for the weakly non-unitary regime and the singular origin regime.