Related papers: General Eigenvalue Correlations for the Real Ginib…
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…
The starting point of this work is a theorem due to Maxwell characterizing the distribution of a Gaussian vector with at least two coordinates. We define the Gaussian orthogonal, unitary and symplectic tensor ensembles for notions of real…
Symplectic ensemble of disordered non-Hermitian Hamiltonians is studied. Starting from a model with an imaginary magnetic field, we derive a proper supermatrix $\sigma $-model. The zero-dimensional version of this model corresponds to a…
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian…
We consider the real eigenvalues of an $(N \times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau_N\in [0,1]$. In the almost-Hermitian regime where $1-\tau_N=\Theta(N^{-1})$, we obtain…
We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}}…
We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B$|$DI) and the ensemble of…
The Ginibre ensemble of complex random Hamiltonian matrices $H$ is considered. Each quantum system described by $H$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. For generic…
We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[\sigma, +\infty[$. In particular,…
We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble to its non-hermitian counterpart - for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green's functions, which…
We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done…
The fermionic Gaussian operator basis provides a representation for treating strongly correlated fermion systems, as well as playing an important role in random matrix theory. We prove that a resolution of unity exists for any even…
We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the…
Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the…
For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion…
The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large $N$ expansions. These topics are inter-related. For example, a third…
We consider the product of \(k_{n}\) independent \(n\times n\) complex Ginibre matrices and denote its eigenvalues by \(Z_{1},\ldots ,Z_{n}\). Let \(\alpha = \lim_{n\to\infty} n / k_{n}\). Using the determinantal point process method, we…
We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors…
Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint…
Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$. When some eigenvalues of $X_0$ separate from the unit disk, outlier…