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,k-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: This talk explores last-passage percolation in the environment partitioned into blocks of varying characteristics. As the number of blocks grows, the last-passage time evolves as a discrete-time stochastic process. I will present an explicit formula for its finite-dimensional distributions, expressed via the Fredholm determinant of a kernel that also appears in the study of singular values of products of random matrices. The model admits a scaling limit in which the last-passage time process converges to the critical stochastic process of random matrix theory. The latter is the top curve in the line ensemble associated to products of matrices under the critical scaling, in which the number of factors and matrix sizes approach infinity in a proportional manner. Time permitting, I will also discuss the relevant hard-to-soft edge transition. The talk is based on recent joint work with Eugene Strahov (HUJI).
Abstract: A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations. The distribution of Bethe roots reveals the presence of a superfluid-Mott insulator transition at the ground state, and the critical point is determined. By adjusting the boundary parameter, we demonstrate the existence of a non-Hermitian skin effect even in the presence of interaction, but it is completely suppressed for the Mott insulator state in the thermodynamical limit. Our result represents a new class of exactly solvable non-Hermitian many-body systems, which has no Hermitian correspondence and can be used as a benchmark for various numerical techniques developed for non-Hermitian many-body systems.
Abstract: Recently, random walks on Hecke algebras were recognized by A. Bufetov as a natural framework for the study of multi-species interacting particle systems. As a corollary, the Mallows measure can be viewed as the universal stationary blocking measure of interacting particle systems arising from random walks on Hecke algebras. Furthermore, the involution in Hecke algebras implies the color-position symmetry, which is a powerful tool for the asymptotic analysis of multi-species interacting particle systems. In this talk, we explore two facets of random walks on Hecke algebras. The first part focuses on the asymptotic behavior of the Mallows measure. In the second part, we consider applications of the color-position symmetry, particularly in the context of shock fluctuations in the half-line open Totally Asymmetric Simple Exclusion Process (TASEP) and Asymmetric Simple Exclusion Process (ASEP).
Abstract: This talk explores the connection between orthogonal polynomials and random matrices, with a focus on the Gaussian Unitary Ensemble (GUE). After introducing key notions and properties of orthogonal polynomials, I will examine how they arise in the study of random matrix models. Building on results by Bleher and Deaño, I will discuss the effects of perturbations of the GUE and consider a model of orthogonal polynomials associated with a cubic potential. In this setting, the free energy admits a topological expansion related to graph enumeration on Riemann surfaces, with coefficients expressed through a solution of the Painlevé I equation. I will also present recent joint work with G. Silva (USP-ICMC) and M. Yattselev (Purdue University) on modifications of the measure in the cubic potential case. Our analysis yields an asymptotic expansion for the recurrence coefficients in inverse powers of N2, revealing a connection with a perturbed Painlevé I equation.
Abstract: We consider the six-vertex model on the N x N lattice, with domain wall boundary conditions, and at ice-point, Δ = 1/2. We focus on the Emptiness Formation Probability (EFP), for which we build an explicit and exact (although still conjectural) expression, as the Fredholm determinant of some linear integral operator. We study the asymptotic behaviour of the obtained representation at large N. In particular, by tuning the geometric parameters of the EFP to the vicinity of the arctic curve, we obtain the GUE Tracy-Widom distribution. Joint work with Andrei Pronko.
Abstract: The β-ensemble is a probabilistic model of N particles on the real line, confined by a potential V and interacting via a logarithmic repulsion. For certain choices of V, this probability measure coincides with the distribution of the spectrum of certain random matrix ensembles. In this talk, we focus on the potential V(x) = |x|p with p ∈ (2, 3). Since V is not of class C3, much of the existing literature—on CLTs, partition function expansions...— does not directly apply. We will demonstrate that a Central Limit Theorem still holds in this singular setting and establish the variance conjecture—a relaxed form of the KLS conjecture in geometric analysis—for a specific class of convex bodies. This is joint work with Ronan Mémin and Michel Pain (both at IMToulouse).
Abstract: The behavior of correlation functions in one-dimensional quantum systems at zero temperature is now very well understood in terms of linear and non-linear Luttinger models. The "microscopic" justification of these models consists in the exact accounting for soft-mode excitations around the vacuum state and, at most few high-energy excitations. At finite temperature, or more generically for finite entropy states, this direct approach is not applicable due to the different structure of "soft" excitations. We focus on physical systems where the strong interaction makes it possible to present correlation functions in terms of Fredholm determinants of the generalized sine kernels. Based on "microscopic" resummations, we develop a phenomenological approach of the effective form factors that allows us to describe the asymptotic behavior of these Fredholm determinants. We demonstrate how this works for correlation functions in the XY model, mobile impurity, and the generic Toeplitz determinants. We explain how this approach is related to the Riemann-Hilbert methods.
Abstract: We introduce a notion of local level spacings and study their statistics within a random-matrixtheory approach. In the limit of infinite-dimensional random matrices, we determine universal sequences of mean local spacings and of their ratios which uniquely identify the global symmetries of a quantum system and its internal—chaotic or regular—dynamics. These findings, which offer a new framework to monitor single- and many-body quantum systems, are corroborated by numerical experiments performed for zeros of the Riemann zeta function, spectra of irrational rectangular billiards, and many-body spectra of the Sachdev-Ye-Kitaev Hamiltonians. [Phys. Rev. Lett. 132, 220401 (2024)].
Abstract: In this talk, we explore random domino tilings of multiply-connected domains and the distinctive features. One key difference is in the variational approach, which leads to two versions of the limit theorem instead of one from Cohn-Kenyon-Propp 01’. As a central example, we study the Aztec diamond with a hole, applying the tangent plane method—a technique for parametrizing limit shapes via harmonic functions, introduced by István Prause and Richard Kenyon in 2020. In the final part of the talk, we turn to parametrization of the limit shape of the Aztec diamond with a hole, where elliptic functions serve as the main tool.
Abstract: Schur measures are probability measures on Young diagrams depending on two countable sets of parameters. Introduced by Okounkov in 2003 and containing the poissonized Plancherel measure as the most known example, they lead to determinantal point processes on the one dimensional lattice. Okounkov also proved that the gap probabilities of these measures are tau functions for the 2D Toda lattice hierarchy, that is to say, they satisfy a hierarchy of particular bilinear PDEs. In this talk, I will show how to extend this result to expectations of more general multiplicative functionals. I will try to give a comprehensive exposition of the techniques of the proof which use the fermionic Fock space formalism. As an application, we recover a recent result of Cafasso- Ruzza on the finite temperature discrete Bessel process, which corresponds to a deformation of the poissonized Plancherel measure, and also obtain a hierarchy for more general finite temperature Schur measures.
Abstract: We consider the elliptic Calogero-Moser system of BCn type with five arbitrary constants and propose R-matrix valued generalization for 2n × 2n Takasaki’s Lax pair. For this purpose we extend the Kirillov’s B-type associative Yang-Baxter equations to the similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa-Ueno R-operator and the Komori-Hikami Koperators satisfying reflection equation. Then, using the Felder-Pasquier construction the answer for the Lax pair is also written in terms of the Baxter’s 8-vertex R-matrix. As a by-product of the constructed Lax pair we also conjecture BCn type generalization for the elliptic XYZ long-range spin chain. (joint work with A. Mostovskii and A. Zotov arXiv:2503.22659).
Abstract: The six-vertex model with domain-wall boundary conditions was introduced by Korepin. It was shown that the partition function of the model satisfies a list of properties, among which is a recursion relation. Later Izergin obtained a determinant formula satisfying these properties for both the rational and trigonometric Boltzmann weights of the model. Recently, another determinant representation for the partition function appeared in the work of Kostov for the rational parametrization, and in the work of Foda and Wheeler for the rational and a special case of trigonometric parametrizations. We formulate a list of properties of the partition function of the six-vertex model on an N × N square lattice with domain wall boundary conditions for both the rational and trigonometric weights, where the recursion relation proposed by Korepin is replaced by a system of algebraic equations with respect to one set of the spectral parameters of the model. We prove that the solution of the system is unique and construct a determinant satisfying all the properties. The resulting determinant depends on an arbitrary basis of polynomials of degree N − 1. By choosing the Lagrange interpolation polynomials we reproduce the original formula by Izergin and by choosing the monomial basis, we derive the representation by Kostov and by Foda and Wheeler including one for generic trigonometric weights. The talk is based on the results of a joint work with A. G. Pronko and V. O. Tarasov.
Abstract: From the general theory of point process, the Laplace functional E[e− R f dξ], where f is in some specific class of functions, give us important information about the process itself. For Determinantal Point Process, the Laplace functional E[e−λ R f dξ] can be written as a Fredholm determinant of an integral operator K. In this talk we investigate the asymptotic expansion of E[e−λ R f dξ] for the Sine process as λ goes to +∞. We demonstrate, via Riemann-Hilbert analysis, that the configurations ξ j ∞ j=1 in which the points are far from the global maximum of f contribute more on the Laplace functional. This result will provide crucial information about the linear statistics and tail probabilities of the Sine process.
Abstract: Understanding and quantifying quantum entanglement in many-body systems remains a central challenge in modern theoretical physics. In this talk, I will present recent results highlighting fruitful connection between orthogonal polynomials and entanglement measures in inhomogeneous quantum systems. Building on tools from the theory of special functions, I will show that certain entanglement characteristics—such as Rényi entropies, entanglement spectra and negativity—can be expressed in terms of classical orthogonal polynomials of the Askey scheme. These results open new avenues for the analytical study of entanglement in a broad class of inhomogeneous models.
Abstract: The connection between edge distributions in random matrix theory and Painlevé equations begun with the discovery of the Tracy-Widom distribution in 1992, which showed that the distribution function of the largest eigenvalue in the GUE was given in terms of a specific solution (Hastings-Mccleod) of the Painlevé-II equation. Since then, much work has gone into showing connections between random matrix models and solutions to different Painlevé equations, specifically work by Forrester and Witte in the early 2000s. In this talk, I will describe some random matrix ensembles with an edge spectrum singularity whose edge distributions admit Painlevé connections, but to different Painlevé equations.
Abstract: We investigate the statistics of particle current fluctuations in the totally asymmetric simple exclusion process (TASEP) on a ring of size N with p particles. By deforming the Markov generator with a parameter γ, we analyze the tilted operator governing current statistics using Bethe ansatz techniques. We derive implicit expressions for the scaled cumulant generating function (SCGF, largest eigenvalue) and spectral gap in terms of Bethe roots, exploiting their geometric structure on Cassini ovals. In the thermodynamic limit N → +∞, we demonstrate a phase transition: the SCGF scales linearly with N for a finite γ > 0, but remains O(1) for γ < 0, reflecting distinct dynamical regimes. The spectral gap, which governs relaxation times, is shown to close algebraically with N, providing insights into metastability in driven systems.
Abstract: A tiling of a hexagon with doubly periodic weights gives rise to a determinantal point process. The correlation kernel of this process has been expressed as a double contour integral by Duits and Kuijlaars. The integrand contains the reproducing kernel associated to matrix-valued orthogonal polynomials. Recently, these polynomials have been analyzed by Kuijlaars for a special case with 3x3-periodic weights. In this case, we determine a scaling limit of the correlation kernel by means of steepest descent analysis. The position of the dominant saddle point determines the phase. There are three phases inside the hexagon: solid, liquid, and gas. We show that the liquid phase is homeomorphic to the (double) amoeba of a genus one Harnack curve. This double amoeba is used to analyze the phase functions and to prove the existence of steepest descent (and ascent) paths. This talk is based on joint work with Arno Kuijlaars.
Abstract: We investigate the t - W scheme for the anti-ferromagnetic XXX spin chain under both periodic and open boundary conditions. We propose a new parametrization of the eigen- values of transfer matrix. Based on it, we obtain the exact solution of the system. By analyzing the distribution of zero roots at the ground state, we obtain the explicit ex- pressions of the eigenfunctions of the transfer matrix and the associated W operator in the thermodynamic limit. We find that the ratio of the quantum determinant with the eigenvalue of W operator for the ground state exhibits exponential decay behavior. Thus this fact ensures that the so-called inversion relation (the t - W relation without the W -term) can be used to study the ground state properties of quantum integrable systems with/without U (1)-symmetry in the thermodynamic limit.