Random matrices close to Hermitian or unitary: overview of methods and results

Abstract

Abstract. The paper discusses recent progress in understanding statistical properties of eigenvalues of (weakly) non-Hermitian and non-unitary random matrices. The first type of ensembles is of the formĴ =Ĥ − iΓ, withĤ being a large random N × N Hermitian matrix with independent entries "deformed" by a certain anti-Hermitian N × N matrix iΓ satisfying in the limit of large dimension N the condition: TrĤ 2 ∝ N TrΓ 2 . HereΓ can be either a random or just a fixed given Hermitian matrix. Ensembles of such a type withΓ ≥ 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics. Related models are used to mimic complex spectra of the Dirac operator with chemical potential in the context of Quantum Chromodynamics.Ensembles of the second type, arising naturally in scattering theory of discrete-time systems, are formed by N × N matrices with complex entries such that †Â =Î −T . ForT = 0 this coincides with the Circular Unitary Ensemble, and 0 ≤T ≤Î describes deviation from unitarity. Our result amounts to answering statistically the following old question: given the singular values of a matrix describe the locus of its eigenvalues.We systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non-trivial crossover from WignerDyson statistics of real/unimodular eigenvalues typical for Hermitian/unitary matrices to Ginibre statistics in the complex plane typical for ensembles with strong non-Hermiticity: < TrĤ 2 >∝< TrΓ 2 > when N → ∞. Finally we discuss (scarce) results available on eigenvector statistics for weakly nonHermitian random matrices. IntroductionAs is well-known, the statistics of highly excited bound states of closed quantum chaotic systems of quite different microscopic nature is universal. Namely, it turns out to be independent of the microscopic details when sampled on energy intervals large in comparison with the mean level spacing ∆, but smaller than the so called Thouless energy scale. The latter is related by the Heisenberg uncertainty principle to the relaxation time necessary for the classically chaotic system to reach equilibrium in phase space [1]. Moreover, the spectral correlation functions turn out to be exactly those which are provided by the theory of large random Hermitian matrices with independent, identically distributed Gaussian entries. The correspondence holds in the limit of large matrix dimension on the so-called local scale. The Random matrices close to Hermitian or unitary2 local scale is determined by the typical separation ∆ = X i − X i−1 between neighbouring eigenvalues situated around a point X, with the brackets standing for the statistical averaging. Microscopic arguments supporting the use of random matrices for describing the universal properties of quantum chaotic systems have been provided recently by several groups, based both on traditional semiclassical periodic orbit expansions [2,3] and on advanced field-theoretical methods [4,5].In paralle...

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

“…Using a saddle-point approximation (the exponent achieves its minimum at the boundary t = 1 of the integration domain) we eventually obtain the same result as in (34) lim…”

“…It would be interesting to see if the results reported in this paper can be extended, at least qualitatively, to the 2D-OCP in more generic confining potentials [14,42,61] (other than radially symmetric [3,34,36]) or for the plasma on different planar surfaces (e.g. on a cylinder [11,12]).…”

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

“…Another important goal is to apply them to ensembles of non-Hermitian random matrices [26,28], which are certain deformations of the invariant class.…”

Rank One Non-Hermitian Perturbations of Hermitian $$\beta $$ β -Ensembles of Random Matrices

“…The asymptotic analysis of the weakly non-Hermitian ensembles are of high interest in the mathematics and physics literature and have been studied in [3,4,6,14], see also [11,12]. The numerous physical applications of such random matrices can be found in the review papers [6,7,10].…”

“…The numerous physical applications of such random matrices can be found in the review papers [6,7,10].…”