A Julia package containing Distributions.jl-type specifications for various distributions arising from random matrix theory.

• SpikedWigner(beta, n, spikes; scaled=false): Wigner distribution with an added spike matrix.

  spikes is an array [s1, ..., sr] such that the diagonal matrix with diagonal sqrt(n)(s₁, ... , s_r, 0, ..., 0) is added to a white Wigner matrix.

• SpikedWishart(beta, n, p, spikes; scaled=false): Wishart distribution with spiked covariance [1].

  spikes is an array [s1, ..., sr] such that the Wishart covariance is diagonal with entries (1 + s₁, ... , 1 + s_r, 1, ..., 1).

• Jacobi(beta, n1, n2, p): Random matrices of the form E(E+H)^-1. Here E and H are (n₁, p) and (n₂, p) white Wisharts respectively. [2]

Specifying scaled=true in SpikedWigner and SpikedWishart scales the matrices by an appropriate function of n so that the corresponding bulks converge to the semicircle and Marchenko-Pastur laws respectively. Due to the inverse in the definition of the Jacobi ensemble, no scaling is necessary for Jacobi,

Normal entries in Gaussian ensembles are scaled to have variance 1.

The package implements the following types:

• MarchenkoPastur(gamma): Limiting empirical spectral density of a real white Wishart matrix with p/n -> gamma as long as 0 < gamma < 1.
• TracyWidom(beta): Limiting distribution of the maximum eigenvalue of many random matrix ensembles with Dyson parameter beta [3].
• Wachter(gamma1, gamma2): Limiting empirical spectral density of S₁ S₂^-1. Here S₁ and S₂ are sample covariance matrices with n₁/p -> gamma₁ and n₂/p -> gamma₂.

It also implements the following functions for computing eigenvalue distributions:

• supercrit_dist(E): Approximate distribution of the supercritical eigenvalues of a matrix drawn from the ensemble E. Currently implemented for Wishart with beta = 1 [4], beta = 2 [5] and Wigner with beta = 1 [6], beta = 2 [7]

The function randeigvals efficiently samples from the distribution of eigenvalues of the implemented random matrix distributions. It does this by generating a tridiagonal or banded matrix with eigenvalue equal in distribution to the specified model.

See the documentation.

[1] Dumitriu & Edelman, "Matrix Models for beta ensembles," Journal of Mathematical Physics, 11 (2002).

[2] Killip & Nenciu, "Matrix Models for Circular Ensembles," International Mathematics Research Notices, 50 (2004).

[3] Bornemann, "On the numerical evaluation of distributions in random matrix theory: a review," (2010).

[4] Baik, Ben Arous & Peche, "Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices," The Annals of Probability, 33 (2005).

[5] Paul, "Asymptotics of sample eigenstructure for a large dimensional spiked covariance model," Statistica Sinica, 17 (2007).

[6] Feral, Peche, "The largest eigenvalue of rank one deformation of large wigner matrices," Commun. Math. Phys., 272 (2007).

[7] Peche, "The largest eigenvalue of small rank perturbations of Hermitian random matrices," Probability Theory and Related Fields, 134 (2006).