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We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…
These notes are based on the lectures delivered at the Les Houches Summer School in July 2015. They are addressed at a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part…
Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices.…
In this short note we address a gaussian property of normal vectors in random non-Hermitian matrices. The approach uses a simple geometric and comparison technique.
We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of…
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…
We report on the development of a systematic variational perturbation theory for the euclidean path integral representation of the density matrix based on new smearing formulas for harmonic correlation functions. As a first application, we…
In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing…
Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-d integrable model with the $1/r^2$ interaction (the Calogero-Sutherland-Moser system), and…
Models of disorder with a direction (constant imaginary vector-potential) are considered. These non-Hermitian models can appear as a result of computation for models of statistical physics using transfer matrix technique or describe…
We review various combinatorial applications of field theoretical and matrix model approaches to equilibrium statistical physics involving the enumeration of fixed and random lattice model configurations. We show how the structures of the…
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
We discuss the problem of adding random matrices, which enable us to study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of ``gluon connectedness," we calculate…
Random matrices have their roots in multivariate analysis in statistics, and since Wigner's pioneering work in 1955, they have been a very important tool in mathematical physics. In functional analysis, random matrices and random structures…
Consider an $n \times n$ non-Hermitian random matrix $M_n$ whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of $f(M_n)$ as $n$ tends to infinity, where…
Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according to the Bures measure. The eigenvalues of…
The nonequilibrium dynamics in chaotic quantum systems denies a fully understanding up to now, even if thermalization in the long-time asymptotic state has been explained by the eigenstate thermalization hypothesis which assumes a universal…
We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our…
We consider random matrix ensembles on the set of Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the…
Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new…