Related papers: A Classification of Non-Hermitian Random Matrices

We study a new class of matrix models, formulated on a lattice. On each site are $N$ states with random energies governed by a Gaussian random matrix Hamiltonian. The states on different sites are coupled randomly. We calculate the density…

Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian…

We axiomatize and study the matrices of type $H\in M_N(A)$, having unitary entries, $H_{ij}\in U(A)$, and whose rows and columns are subject to orthogonality type conditions. Here $A$ can be any $C^*$-algebra, for instance $A=\mathbb C$,…

We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to M. Gromov's well-known constructions and include for example a "density model" for…

We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution,…

We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with…

Recently, S\'a, Ribeiro and Prosen introduced complex spacing ratios to analyze eigenvalue correlations in non-Hermitian systems. At present there are no analytical results for the probability distribution of these ratios in the limit of…

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative…

Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present…

We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace…

A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum…

Non-Hermiticity significantly enriches the concepts of symmetry and topology in physics. Particularly, non-Hermiticity gives rise to the ramified symmetries, where the non-Hermitian Hamiltonian $H$ is transformed to $H^\dagger$. For…

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the…

We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq…

It is well known that an (in general, non-commutative) set of non-Hermitian operators $\Lambda_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus…

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…

This paper expands the analysis of randomized low-rank approximation beyond the Gaussian distribution to four classes of random matrices: (1) independent sub-Gaussian entries, (2) independent sub-Gaussian columns, (3) independent bounded…

We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries…

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian…

The spectral densities of ensembles of non-Hermitian sparse random matrices are analysed using the cavity method. We present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated.…