RandomMatrixDistributions.jl
A Julia package containing =Distribution.jl=type specifications for various distributions arising from random matrix theory.
Currently implemented distributions
Matrix distributions

SpikedWigner(beta, n, spikes; scaled=false)
: Wigner distribution with an added rankr matrix with eigenvalues (s_{1}, ... , s_{r}) * sqrt(n). 
SpikedWishart(beta, n, p, spikes; scaled=false)
: Wishart distribution with spiked covariance (sampler for more than one spike implemented only for the real case. [1]spikes
is an array[s1, ..., sr]
such that the Wishart covariance is diagonal with entries s_{1}, ... , s_{r}, 1, ..., 1. 
Jacobi(beta, n1, n2, p)
: Random matrices of the form E(E+H)^{1}. Here E and H are (n_{1}, p) and (n_{2}, 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 MarchenkoPastur 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.
Limiting eigenvalue distributions
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.Wachter(gamma1, gamma2)
: Limiting empirical spectral density of S_{1} S_{2}^{1}. Here S_{1} and S_{2}$ are sample covariance matrices with n_{1}/p > gamma_{1} and n_{2}/p > gamma_{2}$.
Efficient samplers
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.
Examples
An Jupyter notebook demonstrating all of the implemented eigenvalue samplers is provided in /examples/eigenvaluesimulation.ipynb
.
References
[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).