2025 Seminars

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik–Ben Arous–P´ech´e (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy– Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes. This is joint work with Ji Oon Lee.

Abstract: Two recent results that offer new structural insights into beta models will be presented. The first part of the talk focuses on stochastic domination in Hermite and Laguerre beta ensembles. In the second part, we present log-concavity phenomena in several discrete and continuous Coulomb gas models. This part of the talk is based on joint work with Manjunath Krishnapur and Mokshay Madiman.

Abstract: This work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy elements in the symmetric group algebra. We extend this picture to integration over both complex and real Grassmannians. The large N expansion for such integrals leads naturally to two distinct deformations of the monotone Hurwitz numbers. The first produces a family of polynomials that are conjectured to satisfy remarkable interlacing phenomena. The second motivates a deformation of the Jucys-Murphy elements that conjecturally relate to the family of Jack symmetric functions. This talk is based on joint work with Xavier Coulter and Ellena Moskovsky [arXiv:2308.04015, arXiv:2506.04002].

Abstract: We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure $d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z| < 1}d^{2}z,$ where $d^{2}z$ is the two dimensional area measure, $\alpha$ is a parameter that can grow with $n$, while $\gamma>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of truncated unitary matrix ensembles and generalizes the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem.

 This is joint work with Alfredo Deaño, Ken McLaughlin and Nick Simm.

Abstract: TBA

Abstract: The goal of this talk is to tie products of random matrices with a particular last-passage percolation problem in layered environment. It will turn out that finite-dimensional distributions of the last-passage time process can be expressed in terms of a kernel that also appears in the study of products of random matrices. We will present a new scaling limit that yields a multi-time analog of the critical kernel discovered by D.-Z. Liu, D. Wang, and Y. Wang, and later in a slightly different form, by G. Akemann, Z. Burda, and M. Kieburg. The talk is based on joint work with Evgeny Strahov.

Abstract: We study the asymptotic behavior of a class of planar orthogonal polynomials. The weight function is supported on the whole complex plane. This is motivated from two macroscopic point insertions in the spherical ensemble. We derive strong asymptotics for the polynomials, their weighted L^2-norms, and the limiting distribution of their zeros.

 These results correspond to the pre-critical regime of an associated two-dimensional Coulomb gas model, in which the support of the equilibrium measure remains simply connected. Our approach hinges on identifying the mother body of the potential theoretic problem and reformulating the planar orthogonality condition as a non-Hermitian contour orthogonality. This key observation enables a Deift-Zhou steepest descent analysis for a 2 \times2 matrix Riemann-Hilbert problem.

 This is a joint work with Sung-Soo Byun, Peter Forrester and Arno Kuijlaars.

Abstract: A method is presented for calculating the large N asymptotics of the even moments of the averaged characteristic polynomial for the classical beta ensembles with beta even. This reiles on duality formulas, and then proceeds via steepest decent for multiple integrals of fixed dimension. It gives beta generalisations of Fisher-Hartwig asymptotics for Toeplitz determinants with one zero singularity in its generating function. Of particular interest is the form of the constant term, which for the CUE has applications to properites of the Riemann zeta function on the critical line.

Abstract: Let c_k be independent standard complex normals. This talk is about zeros of Gaussian Analytic Functions f(z) = \sum c_k z^k, conditioned by the event that f(0)=a. The probability law of the zero set of f(z) can be derived from that of spectral outliers of random sub-unitary matrices. I will explain how this link can be used to obtain the full conditional distribution of radial zero counting function, prove its asymptotic normality and develop precise asymptotic form of the probabilities of moderate to large deviations from normality. One also finds tail probabilities for the conditional distribution of the k-th smallest absolute value of zeros for which the probability model is one of the order statistics of infinite sequence of independent random variables satisfying a global constraint. This model exhibits interesting features. For instance, the asymptotic expansion for the conditional hole probability has two built-in two scales (slow and fast) and, to leading order the conditional hole probability coincides with that of the unconditioned GAF of the form \sum \sqrt{k+1} c_k z^k. My talk is based on joint work with Yan Fyodorov and Thomas Prellberg (arXiv:2412.06086 [1]).

Abstract: This talk explores the limiting empirical zero distributions of polynomials as their degree tends to infinity. In particular, we discuss the role of the coefficients, the implications for (finite) free probability and how zeros evolve under certain differential flows.

 First, we shall focus on real-rooted polynomials and present a user-friendly approach to determine real limit zero distributions via the 'exponential profile' of the coefficients. Apart from applications to classical polynomial ensembles, this enables us to study the effect of repeated differentiation, the heat flow and finite free convolutions (which converges to the free convolutions even for non-compactly supported distributions).

 Second, we discuss random polynomials with i.i.d. rescaled coefficients and the evolution of their (complex) zeros under certain differential flows. For instance, the limiting zero distribution of Weyl polynomials undergoing the heat flow evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law. 

 We interpret this from the perspective of free probability, optimal transport, and PDE's, and accompany the results by illustrative simulations.

Based on joint works with Brian Hall, Ching-Wei Ho, Antonia Höfert, Zakhar Kabluchko, and Alexander Marynych.

Abstract: The asymmetric Wishart ensemble, introduced by Akemann, Phillips, and Sommers, interpolates between the chiral real Ginibre ensemble and the Wishart (Laguerre) orthogonal ensemble via a one-parameter family.

Although significant progress has been made in recent years on the study of real eigenvalues in the Ginibre orthogonal ensemble, the elliptic Ginibre orthogonal ensemble, and products of Ginibre orthogonal matrices, the rigorous computation of the expected number of real eigenvalues in the asymmetric Wishart ensemble has remained an open problem.

 One reason for this is the lack of an integrable formula that decomposes the correlation function into a complex counterpart (DPP-correlation kernel) and a rank-one perturbation term.

 In this talk, we introduce a novel integrable formula that enables the analysis of the statistical properties of real eigenvalues in the asymmetric Wishart ensemble, and we address the expected number of real eigenvalues of the asymmetric Wishart ensemble. This talk is based on a joint work [arXiv:2503.14942] with Sung-Soo Byun (Seoul National University).

Abstract: In the large N limit, random matrix models exhibit limiting spectra in the complex plane whose support is called the droplets. In this talk, we will discuss the elliptic Ginibre matrix model conditioned to have real eigenvalue with multiplicity proportional to the dimension of the matrix. We prove that the droplets are either simply connected, doubly connected, or composed of two simply connected components. Moreover, we present the explicit description of the droplet and electrostatic energies for the simply and doubly connected case. Finally, we introduce the asymptotic behavior of the moments of the characteristic polynomials of elliptic Ginibre matrices as an application. This is based on a joint work with Sung-Soo Byun. (arXiv:2502.02948)

Abstract: I will talk about the two related results mentioned in the title, both of them concern the Wigner minor process, which is a sequence of appropriately scaled $N\times N$ upper left corners of a doubly infinite symmetric array of i.i.d. random variables. We establish the analogue of the Hartman-Wintner law of iterated logarithm for the top eigenvalue of these matrices. This result was initially coined as a law of fractional logarithm (LFL) by E. Paquette and O. Zeitouni, who resolved the special case of GUE matrices. Our work verifies the 10-year-old conjecture by these authors, proving the LFL in full generality for both symmetry classes. Additionally, we show that the correlation between the top eigenvalues in the minor process becomes weaker as the difference between the sizes of the minors increases. We establish the precise description of the resulting decorrelation transition, extending the result of J. Forrester and T. Nagao for the GUE case. The talk is based on the recent joint works with Z. Bao, G. Cipolloni, L. Erd{\H o}s and J. Henheik.

Abstract: I will present a new method to characterize gap probabilities of discrete determinantal point processes in terms of Riemann-Hilbert problems. Simple examples of such discrete point processes arise in domino tilings of Aztec diamonds and lozenge tilings of hexagons. As a first illustration of our approach, we obtain a new explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Padé approximants, by solving the associated Riemann-Hilbert problem. As a second application, we obtain an explicit expression for the number of lozenge tilings of reduced hexagons in terms of Hermite-Padé approximants. 

This is based on joint work with Christophe Charlier.

Abstract: Mesoscopic fluctuations for orthogonal polynomial ensembles have been studied in the literature since 2000. However, most works focus on the bulk, and less is known about the edges. In this talk, I am reporting on our recent work on the mesoscopic edge universality of orthogonal polynomial ensembles arXiv:2501.14422. I will present the main theorems and the ideas of the proof with an example of Laguerre Unitary Ensemble (LUE). The interesting aspect of LUE is that it presents both hard and soft edges. Our approach reveals how the different types of edges of LUE affect the mesoscopic fluctuations. Universality comes naturally from our approach. The main tool used in this study is the three-term recurrence relation of the associated orthogonal polynomials.

Abstract: In their 2001 paper, Forrester-Witte showed, using Okamoto's tau-function theory, that the generating functional of the GUE with an edge spectrum singularity of order |z-√2|^α is related to the σ-form of the Painlevé-II equation with coefficient α, generalising the result that was known for the edge of spectrum distribution for the GUE. We use a method of orthogonal polynomials to generalise this result to a class unitary ensembles with more general weight functions exp(-n tr(V(M)). In this talk I will introduce some background material on these ensembles and demonstrate firstly how we can obtain all of the asymptotic information for such ensembles by simply considering the large-n asymptotics of the relevant orthogonal polynomials.  

Abstract: TBA

Abstract: The purpose of this talk is to understand the limiting behavior of empirical root distributions of polynomials when the degree tends to infinity in the framework of free probability. More precisely, we present how to relate the asymptotic behavior of the ratio of consecutive coefficients (namely, finite S-transform) of polynomials and Voiculescu’s S-transform (from free probability) of limiting empirical root distribution of the polynomials as the degree tends to infinity. This is a joint work with Octavio Arizmendi, Katsunori Fujie and Daniel Perales.

Abstract: I will discuss some recent results and techniques related to the extremes of characteristic polynomials. I will first discuss the case of the C$\beta$E, where techniques based on orthogonal polynomials, first exploited in this context by Chhaibi, Madaule and Najnudel, lead to an answer to the Fyodorov-Hiary-Keating prediction for CUE characteristic polynomials. I will then describe (partial) results for Hermitian matrices.

Abstract: The characteristic polynomials of random matrices have long been studied due their complex behavior, which reflects the behavior of the eigenvalues themselves, and the number of applications they arise in. However, despite considerable work, relatively few results are known about these characteristic polynomials, due to the intractability of computing its statistics in all but the nicest ensembles. In particular, results are primarily known only for symmetric/Hermitian and unitary matrices. In this talk, we discuss work computing the moments of the absolute characteristic polynomial for a particular family of non-symmetric matrix ensembles, namely the real elliptic ensemble. Traditionally these computations employ a super-symmetric or combinatorial method, while our method is based on a relation between correlations of characteristic polynomials and the correlation functions of larger matrices. We also discuss our application of this result to compute the number of equilibria in the asymmetric spherical p-spin glass, a non-gradient dynamical system. In particular, we apply our moment computation to demonstrate a conjectured relative to absolute instability transition in the model.

Abstract: A Beta-ensemble can be viewed as a gas of particles confined to a line with logarithmic pairwise interactions and at inverse temperature Beta. The partition function of a Beta-ensemble involves a large parameter N which appears both in the integrand and as the number of integrations, and thus its asymptotic analysis could be regarded as an infinite-dimensional version of the Laplace method. In joint work with A. Guionnet and K. Kozlowski, we consider the partition function of a Beta-ensemble with a complex-valued potential. We prove, under certain hypotheses, a full 1/N expansion of this partition function and explicitly identify the first few terms. Because the integrand is complex, and hence oscillatory, our method could be regarded as an infinite-dimensional version of the Steepest Descent method. ArXiv reference: https://arxiv.org/abs/2411.10610

Abstract: Symmetry of non-Hermitian matrices underpins many physical phenomena. In particular, chaotic open quantum systems exhibit universal bulk spectral correlations classified on the basis of time-reversal symmetry$^\dagger$ (TRS$^\dagger$), coinciding with those of non-Hermitian random matrices in the same symmetry class. In my talk I will present our results on spectral statistics of non-Hermitian random matrices in the presence of TRS$^\dagger$ with signs +1 and −1, corresponding to symmetry classes AI$^\dagger$ and AII$^\dagger$, respectively. Using the fermionic replica non-linear sigma model approach, we derive n-fold integral expressions for the nth moment of the one-point and two-point characteristic polynomials. Performing the replica limit n→0, we qualitatively reproduce the density of states and level-level correlations of non-Hermitian random matrices with TRS$^\dagger$.

Abstract: We consider two-dimensional Coulomb systems in the regime when the droplet is connected, while the coincidence set for the obstacle problem contains an analytic Jordan curve outside of the droplet. A nontrivial (Heine-distributed) number of particles will tend to fall in the vicinity of this curve, which we denote "spectral outpost". Under the process of Laplacian growth, the outpost grows into a new ring-shaped component of the droplet, and the study of outposts is therefore closely related to the regime of disconnected droplets. We study among other things fluctuations of linear statistics. The talk is based on joint works with Joakim Cronvall and Christophe Charlier. 

Abstract: Consider a random matrix X with independent, identically distributed entries, and a deterministic deformation A. We prove that the eigenvalues statistics of A+X are universal close to the edges of its spectrum. Under mild assumptions on A, we also show that A+X does not have outliers at a distance larger than the fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is a deterministic integer.

Abstract: It is well-known that the LUE (LOE) emerges as the reduced density matrices of random (time-reversal symmetric) pure states, which can be obtained by applying random unitaries (orthogonal matrices) to a reference state (represented by a real vector) [1]. In this seminar, we talk about how to realize the remaining LSE, which does not admit a simple analogue [2]. The missing piece of puzzle turns out to be "symmetry fractionalization”, a central concept underlying symmetry-protected topological phases [3]. We further generalize the scenario to arbitrary (finite-group) symmetries that may be “fractionalized". The corresponding reduced density matrices are found to be some direct sums of LOE, LUE and/or LSE. Our result may be considered as a variant of Dyson’s threefold way for quantum entanglement [4]. 

[1] J. Sánchez-Ruiz, Phys. Rev. E 52, 5653 (1995); P. J. Forrester, Log-Gases and Random Matrices (Princeton University Press, 2010).

[2] H. Yagi, K. Mochizuki, and Z. Gong, arXiv:2410.11309.

[3] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983); X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 84, 235128 (2011).

[4] F. J. Dyson, J. Math. Phys. 3, 1199 (1962).

Abstract: We consider ‘λ-shaped random matrices’, whose entries are i.i.d. in the boxes of a given Young diagram λ and zero elsewhere. In particular, we study their limiting spectral distribution when the shape λ is dilated by a growing factor N. The moments of such a distribution are a generalisation of Catalan numbers, and enumerate combinatorial objects which we call λ-plane trees: these are trees whose vertices are labelled in a way that is ‘compatible’ with λ. Based on joint works with Elia Bisi and Marilena Ligabò.

Abstract: In the first part of the talk I will discuss the complex eigenvalue statistics of a XXZ spin chain with imaginary disorder, where we find an interplay between Hermiticity and integrability breaking at different scales of the disorder strength. We compare with the symmetry class of AI† and Poisson statistics in 1 ≤ D ≤ 2 dimension. In particular, the nearest- and next-to-nearest-neighbour spacing distributions of classes AI† and AII† are very well approximated by a two-dimensional Coulomb gas at β = 1.4 and 2.6, respectively. In the second part of the talk, I will turn to the general study of the local bulk statistics of non-Hermitian random matrices. Based on numerically generated nearest-neighbour spacing distributions, it is conjectured that among all 38 non-Hermitian symmetry classes, only 3 different local bulk statistics exist. The simplest representatives are complex Ginibre matrices (class A), complex symmetric matrices (class AI†) and complex self-dual matrices (class AII†). While class A is well understood, only very few is known about the latter two classes. I will present the first analytic results for the expectation value of two characteristic polynomials in classes AI† and AII†. This includes results at finite matrix size as well as global and local edge and bulk asymptotics.

Abstract: I will give a classification of biorthogonal ensembles that are both a multiple orthogonal polynomial ensemble and a polynomial ensemble of derivative type (also called a Pólya ensemble). The notion of polynomial ensemble of derivative type was first introduced by Kieburg-Kösters (2016) because it provided a natural framework to describe the squared singular values of products of random matrices. The analogous notion for the eigenvalues of sums of random matrices was provided later by Kuijlaars-Róman (2019). I will explain how, in these settings, the additional structure of a multiple orthogonal polynomial ensemble leads to a full classification of such ensembles. As a consequence, we obtain families of multiple orthogonal polynomials that decompose naturally using the finite free multiplicative and additive convolution from finite free probability.

Abstract: In studying higher-order finite-size corrections to various limit laws at the soft edge, we discovered a particularly simple yet perplexing structure: the correction terms are linear combinations of higher-order derivatives of the limit law with rational polynomial coefficients. Given the highly nonlinear nature of perturbations of operator determinants, such a result is unexpected and points to a further layer of integrability. Here, it is the unique solvability of certain rectangular linear systems over the ring of rational polynomials with the solution of Painlevé II and its derivative added as indeterminates. We will present the ideas and evidence in the simplest example, the large matrix limit of Gaussian ensembles.