### Article

## Local laws for non-Hermitian random matrices and their products

We consider products of independent \(n \times n\) non-Hermitian random matrices \(\X^{(1)}, \ldots, \X^{(m)}\). Assume that their entries, \(X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, \ldots, m\), are independent identically distributed random variables with zero mean, unit variance. G\"otze -- Tikhomirov~\cite{GotTikh2011} and O'Rourke--Sochnikov~\cite{Soshnikov2011} proved that under these assumptions the empirical spectral distribution (ESD) of \(X^{(1)} \cdots X^{(m)}\) converges to the limiting distribution which coincides with the distribution of the \(m\)-th power of random variable uniformly distributed in the unit circle. In the current paper we provide a local vesion of this result. More precisely, assuming additionally that \(\E |X_{11}^{(q)}|^{4+\delta} < \infty\) for some \(\delta > 0\), we prove that ESD of \(X^{(1)} \cdots X^{(m)}\) converges to the limiting distribution on the optimal scale up to \(n^{-1+2a}, 0 < a < 1/2\) (up to some logarithmic factor). Our results generalize the recent results of Bourgade--Yau--Yin~\cite{Bourgade2014a}, Tao--Vu~\cite{TaoVu2015a} and Nemish~\cite{nemish2017}. We also give further development of Stein's type approach to estimate the the Stieltjes transform of ESD.

I show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos.

We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the HermitePadé approximants. Solutions of such problems can be expressed bymeans of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero. Using the coefficients of the equation we compute the extremal cuts of the Riemann surfaces. These cuts are attractive sets for the poles of the HermitePadé approximants. We demonstrate the method by an example of the equilibrium problem related to a special system that is called the Angelesco system.

Random matrix theory (RMT) is applied to investigate the cross-correlation matrix of a financial time series in four different stock markets: Russian, American, German, and Chinese. The deviations of distribution of eigenvalues of market correlation matrix from RMT global regime are investigated. Specific properties of each market are observed and discussed.

In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function, discontinuous in 0, Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants. Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function.

To a $N \times N$ real symmetric matrix Kerov assigns a piecewise linear function whose local minima are the eigenvalues of this matrix and whose local maxima are the eigenvalues of its $(N-1) \times (N-1)$ submatrix. We study the scaling limit of Kerov's piecewise linear functions for Wigner and Wishart matrices. For Wigner matrices the scaling limit is given by the Verhik-Kerov-Logan-Shepp curve which is known from asymptotic representation theory. For Wishart matrices the scaling limit is also explicitly found, and we explain its relation to the Marchenko-Pastur limit spectral law.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.