Quantifying Dip–Ramp–Plateau for the Laguerre Unitary Ensemble Structure Function

The Distribution of the First Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble

Abstract. The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the La...

On the condition number of the critically-scaled Laguerre Unitary Ensemble

We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX∗, where X is an N × n matrix with iid standard complex normal entries. Under the scaling n = N + b √ 4cNc, c > 0 and N →∞, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices A are all given by the Tracy–Widom distribution ...

Global and Local Limit Laws for Eigenvalues of the Gaussian Unitary Ensemble and the Wishart Ensemble

The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices

We solve the complex extension of the chiral Gaussian Symplectic Ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane and we prove their orthogonality. Alternatively, a complex eigenvalue representation of this ensemble is given for general weight functions. All k-point correl...

Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Consider a n × n matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets (∆i,n, 1 ≤ i ≤ p), properly rescaled, and eventually included in any neighbourhood of the support of Wigner’s semi-circle law, we prove that the related counting measures (Nn(∆i,n), 1 ≤ i ≤ p), where Nn(∆) represents the number of eigenvalues within ∆, are asymptotical...