Library

EigvalDist(matrixdist::MatrixDistribution)

The joint distribution of eigenvalues sampled from a random matrix with the distribution matrixdist.

Jacobi(β::Int, n₁::Int, n₂::Int, p::Int)

Distribution of a p×p Jacobi matrix.

If E ~ Wishartₚ(I, n₁) and H ~ Wishartₚ(I, n₂) are independent with Dyson parameter β, then E(E + H)⁻¹ has Jacobi(β, n₁, n₂, p) distribution.

If λᵢ are the eigenvalues of EH⁻¹ and μᵢ are the Jacobi eigenvalues, then μᵢ = λᵢ/(1 + λᵢ) and λᵢ = μᵢ/(1 - μᵢ).

SpikedWigner(β::Int, n::Int[, spikes::Vector{Float64}, scaled::Bool=false])

Distribution on an n×n spiked Gaussian Wigner matrix.

Wigner matrices are Hermitian with independent real, complex or quaternion standard Gaussian entries depending on whether β = 1, 2 or 4.

A diagonal matrix with entries given by spikes multiplied by √n is added to produce the spiked Wigner matrix.

If scaled == true, then the resulting matrix is divided by √n so that its bulk distribution converges to the semicircle law supported on [-2, 2].

SpikedWishart(β::Int, n::Int, p::Int[, spikes::Vector{Float64}, scaled::Bool=false)

Distribution of a p×p spiked Wishart matrix.

If X is a p×n matrix with independent real, complex or quaternion standard Gaussian entries, depending on whether β = 1, 2 or 4, then XX† has a Wishart(β, n, p) distribution.

If Λ is a diagonal matrix whose entries are √(1 .+ spikes), then ΛXX†Λ has a SpikedWishart(β, n, p, spikes) distribution.

If scaled == true, then the resulting matrix is divided by p so that its bulk distribution converges to the Marchenko-Pastur law.

bulk_dist(d::Union{MatrixDistribution, EigvalDist})

Compute the limiting spectral distribution of d.

randeigvals([rng::AbstractRNG, ]d::MatrixDistribution)

Sample a vector of eigenvalues of a matrix drawn from the matrix ensemble d.

randeigstat([rng::AbstractRNG, ]d::MatrixDistribution, eigstat::Function, n::Int)

Sample n realisations of the eigenvalue statistic eigstat evaluated at a matrices drawn from the ensemble d.

eigstat is a function of a square matrix argument whose value depends only on the eigenvalues of that matrix.

Usage

rangeigstat(SpikedWigner(2, 50), eigmax, 100)

supercrit_dist(d::Union{MatrixDistribution, EigvalDist})

Compute the approximate joint distribution of the supercritical eigenvalues of the ensemble d.

MarchenkoPastur(γ::Real)

Marchenko-Pastur distribution, where 0 < γ ≤ 1.

The limiting spectral distribution of a p×p covariance matrix of n standard normal observations, where p/n → γ.

TracyWidom(β::Int)

Tracy-Widom distribution with Dyson parameter β.

The limiting distribution of the largest eigenvalue of a GOE (β = 1), GUE (β = 2) or GSE (β = 4) matrix.

Wachter(γ₁::Real, γ₂::Real)

Wachter distribution, where 0 ≤ γ₂ < 1.

Let Σ₁ and Σ₂ be p×p covariance matrices of n₁ and n₂ standard normal observations respectively.

If p/n₁ → γ₁ and p/n₂ → γ₂, then Σ₁ Σ₂⁻¹ has a limiting spectral distribution of Wachter(γ₁, γ₂).