ICMU Winter School
RANDOM MATRICES, RANDOM ANALYTIC
FUNCTIONS, AND NON-LINEAR PDEs
Chynadiyeve village, Zakarpattia, Ukraine
13-19 January 2025
Abstracts of minicourses
Yan V Fyodorov
King’s College London

Statistics of eigenvectors of non-Hermitian randommatrices
A (square) matrix X is called non-normal if it does not commute with its Hermitianadjoint: XX ∗ ̸= X ∗ X. Generically, non-Hermitian random matrices are non-normal. To
each eigenvalue of a non-normal matrix λi , real or complex, correspond two eigenvectors:
left li and right ri . The corresponding eigenproblems are
Xri = λi ri and X ∗ li = λi li .
The two sets can always be chosen bi-orthogonal: (l∗i rj ) = δij .
Consider now an (additively) perturbed matrix X ′ = X + ϵP , with ϵ > 0 controllingthe magnitude of the perturbation P . To the leading order in ϵ the eigenvalues are shiftedbyp|λi (ϵ) − λi (0)| = ϵ|l∗i P ri | ≤ ϵ||P ||2 (l∗i li )(r∗i ri ),showing that the sensitivity of eigenvalues is mainly controlled by the eigenvalue condition numbers:p
κi = (l∗i li )(r∗i ri ) ≥ 1,with κ = 1 only when X is normal. Thus, non-normal matrices are much more sensitive
to perturbations of the matrix entries than their normal counterparts. Moreover, for big
random matrices this sensitivity may become parametrically strong. Further, the condition numbers in random case turn out to be broadly distributed.
The goal of the course will be to provide a detailed discussion of statistics of condition numbers, as well as some related quantities, for the large N × N random matrices
1from the simplest complex Ginibre ensemble, where entries are i.i.d. complex normals:ℜXj,k ∼ ℑXj,k ∼ N −1/2 N (0, 1/2). The technique employed will be a combination of the incomplete Schur decomposition and a variant of the so-called supersymmetry method which uses
integration over anticommuting variables. The discussion of these techniques, especially in thelimit of large N , will be presented at an (in)formal level, with the emphasis on underlying ideasrather than on the rigour of the presentation.

Leonid Pastur
ILTPE, Kharkiv / King’s College London

Large sample covariance matrices and their application to deep learning networks and quantum gravity
Large empirical covariance matrices are the object of active research in a number of areas ofmodern mathematics and related fields. They originated in the 1930s in classical mathematicalstatistics, whose goal was to statistically predict the values of a finite number p of unknown
parameters from the results of a large number of n → ∞ independent measurements (samples),that is, to operate with covariance matrices of size p, in particular, with their eigenvalues as
n → ∞. However, with the advent of the era of big data, when p can be as large as n, it
became clear that classical methods can lead to erroneous predictions, and so the problemof studying large covariance matrices, in particular those where p → ∞, n → ∞, p/n →c ∈ (0, ∞) arose. In the 1950s-1960s, large random matrices of a somewhat different typeappeared in theoretical physics. This gave rise to the theory of random matrices, which has
been actively developed ever since. The proposed course will present a simple and general
method to obtain the eigenvalue distributions of empirical covariance matrices in the above
double-scaling limit. The method is based on elementary facts of linear algebra and probability
theory. A brief overview of applications of these and other large random matrices in theoreticalphysics, quantum information theory, and neural networks will also be given.

Mariya Shcherbina
B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv (ILTPE)

Supersymmetric approach in modern mathematics
Since the end of the last century, methods of supersymmetry, based on the simultaneoususe of commuting and anti-commuting variables, have been widely used in modern theoreticalphysics. The methods proved to be very efficient in analysis of disordered (random) systems,
allowing researchers to obtain many interesting results.
However, an active application of the supersymmetric methods in mathematics is still in
its early stages, although the number of examples of such applications grows every year. The
approach allows researchers to solve many interesting long-standing problems by using thenumber of algebraic tricks that are very similar to the standard linear algebra and Gaussian
integration methods, although they involve the operations with anti-commuting variables. As
it was written in the preface to Efetov’s book on supersymmetry: “All supersymmetric theories
are based on the use of anti-commuting variables introduced by Grassmann in the last century.
At the first glance these objects look very artificial and seem to have no relation to the realworld. There is a certain threshold to start using Grassmann anti-commuting variables in thecalculations because one expects that they should be very unusual. Surprisingly, it is not true2

and, provided proper definitions are given, one can simply generalize conventional mathematicalconstructions so that it is possible to treat both commuting and anti-commuting variables onan equal footing”.
The course aims to give an introduction into the mathematical foundations of supersymmetric approach. It includes the definition of anti-commuting Grassamann variables, super vectors,
super matrices, and operations with them. We introduce integration over Grassmann variables
and compute some important integrals. Finally, some applications to random matrix theory
will be presented.
Necessary prerequisites:• basics of linear algebra,• basics of probability theory.

Dmitry Shepelsky
ILTPE, Kharkiv

Integrable nonlinear partial differential equations andRiemann-Hilbert problems
Riemann-Hilbert (RH) problems are boundary-value problems for (piece-wise) analytic functions in the complex plane. It is a remarkable fact that a vast array of problems in mathematics,
mathematical physics, and applied mathematics can be posed as Riemann-Hilbert problems.
These include radiation, elasticity, hydrodynamic, diffraction problems, orthogonal polynomials
and random matrix theory, nonlinear ordinary and partial differential equations. In applications, the data for a RH problem depend on external parameters, which turns out to be physicalvariables (space, time, matrix size, etc) in the relevant equations modeling particular processes.
Particularly, we are interested in the application of the RH problem approach to a class ofnonlinear partial differential equations (PDE) called “integrable equations” or “soliton equations”. This class of PDE has been the subject of intensive studies since 1970th, when theso-called Inverse Scattering Transform (IST) method was invented. The RH problem approach
can be viewed as a particular realization of the IST method. The representation of a solution
to a nonlinear partial differential equation in terms of a solution of the associated RH problem
can be viewed as a nonlinear analogue of the contour integral representation for a linear PDE.
Particularly, it provides effective means to evaluate accurately the solutions in various asymptotic regimes. The lectures aim at introducing the Inverse Scattering Transform method in theform of a Riemann-Hilbert problem for studying integrable nonlinear differential equations andillustrating the fruitfulness of the method by studying the long-time asymptotics of solutions ofsuch equations. As a prototype model, we use the nonlinear Schrödinger equation, which is abasic model of nonlinear wave propagation in various situations.
Necessary prerequisites:• basics of the complex analysis;• basics of the theory of ordinary differential equations.

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Mikhail Sodin
Tel Aviv University

Random zero sets
Random polynomials and analytic functions have attractedthe attention of mathematicians
for a long time, though the focus of interest has changed over time. Just as the distributionof eigenvalues is central to random matrix theory, the zero sets are central to the study ofrandom analytic functions. In recent decades, the subject has been revived due to numerous
new links to mathematical physics, probability theory, complex and harmonic analysis, andcomplex geometry. In this mini-course, I will try to provide a glimpse into classical and recent
results and open questions pertaining to the zero sets of random polynomials and random
analytic functions.

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