Related papers: How much can the eigenvalues of a random Hermitian…
The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ is deterministic, symmetric with large operator norm and $N_n$ is a random symmetric matrix with…
We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…
We introduce a measure of discrepancy of Hermitian matrices and establish an inequality between the second singular value of a Hermitian matrix and its discrepancy. These results are applied to answer two questions of Fan Chung about graph…
We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional…
We establish a large deviation principle for the largest eigenvalue of a rank one deformation of a matrix from the GUE or GOE. As a corollary, we get another proof of the phenomenon, well-known in learning theory and finance, that the…
We investigate a random normal matrix model with eigenvalues forced to be in the droplet, the support of the equilibrium measure associated with an external field. For radially symmetric external fields, we show that the fluctuations of the…
We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of…
Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem.In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf)…
We consider quadratic forms of deterministic matrices $A$ evaluated at the random eigenvectors of a large $N \times N$ GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove…
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they…
This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical…
A recently developed wavelet based approach is employed to characterize the scaling behavior of spectral fluctuations of random matrix ensembles, as well as complex atomic systems. Our study clearly reveals anti-persistent behavior and…
Symmetric tridiagonal matrices appear ubiquitously in mathematical physics, serving as the matrix representation of discrete random Schr\"odinger operators. In this work we investigate the top eigenvalue of these matrices in the large…
In this paper, we analyze the large n-limit for random matrix with external source with three distinct eigenvalues. And we confine ourselves in the Hermite case and the three distinct eigenvalues are $-a,0,a$. For the case $a^2>3$, we…
These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we…
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for…
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…
Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or, well-known number sequences may display strong signatures that the Hamiltonian yielding them as eigenvalues is…
We consider two non-Gaussian ensembles of large Hermitian random matrices with strong level confinement and show that near the soft edge of the spectrum both scaled density of states and eigenvalue correlations follow so-called Airy laws…