Related papers: On collision of multiple eigenvalues for matrix-va…

We review the state of the art of the theory of Euclidean random matrices, focusing on the density of their eigenvalues. Both Hermitian and non-Hermitian matrices are considered and links with simpler, standard random matrix ensembles are…

Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not…

We calculate the eigenvalues of some two-dimensional non-Hermitian Hamiltonians by means of a pseudospectral method and straightforward diagonalization of the Hamiltonian matrix in a suitable basis set. Both sets of results agree remarkably…

For the eigenvalues of $p$ complex hermitian $n\times n$ matrices coupled in a chain, we give a method of calculating the spacing functions. This is a generalization of the one matrix case which has been known for a long time.

We consider the problem of aligning a pair of databases with jointly Gaussian features. We consider two algorithms, complete database alignment via MAP estimation among all possible database alignments, and partial alignment via a…

We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary…

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth…

So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: $E^n_{\pm}(\lambda)$ with $n=0,1,2...$.…

We obtain an optimal bound for a Gaussian approximation of a large class of vector-valued random processes. Our results provide a substantial generalization of earlier results that assume independence and/or stationarity. Based on the decay…

We derive a sufficient condition for a Hermitian $N \times N$ matrix $A$ to have at least $m$ eigenvalues (counting multiplicities) in the interval $(-\epsilon, \epsilon)$. This condition is expressed in terms of the existence of a…

In this article, for some $d-$dimensional Gaussian processes \[X=\big\{X_t=(X^1_t,\cdots,X^d_t):t\ge0\big\},\] whose components are i.i.d. $1-$dimensional self-similar Gaussian process with Hurst index $H\in(0,1)$, we consider the…

We analyze the convergence to equilibrium in a family of Kac-like kinetic equations in multiple space dimensions. These equations describe the change of the velocity distribution in a spatially homogeneous gas due to binary collisions…

The probability that there are $k$ real eigenvalues for an $n$ dimensional real random matrix is known. Here we study this for the case of products of independent random matrices. Relating the problem of the probability that the product of…

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…

The process of collision of two parallel domain walls in a supersymmetric model is studied both in effective Lagrangian approximation and by numerical solving of the exact classical field problem. For small initial velocities we find that…

In this study, we introduce a two dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time independent Schr\"odinger equation yields real eigenvalues with complex eigenfunctions. We…

We develop several methods, based on the geometric relationship between the eigenspaces of a matrix and its adjoint, for determining whether a square matrix having distinct eigenvalues is unitarily equivalent to a complex symmetric matrix.…

Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an…

In this paper we propose the method to find the hitting probabilities for Gaussian integrators. Using second quantization we obtain the sseries representation for such probabilities despite the fact that integrators can be non-Markov…

We consider a class of $d$-dimensional stochastic differential equations that model a non-colliding random particle system. We provide a sufficient condition, which does not depend on the dimension $d$, for the existence of negative moments…