freealg.distributions.FreeLevy.matrix

FreeLevy.matrix(size, seed=None)

Generate matrix with the spectral density of the distribution.

Parameters:

sizeint

Matrix size n.

seedint, default=None

RNG seed.

Returns:

Anumpy.ndarray

Symmetric matrix (n x n).

Notes

Use a sum of independent (rotationally invariant) Wishart terms, which are asymptotically free:

\[A = \sum_{i=1}^r s_i \frac{1}{m_i} \mathbf{Z}_i \mathbf{Z}_i^{\intercal},\]

where \(\mathbf{Z}_i\) are \(n \times m_i\) i.i.d. \(N(0,1)\). Choose aspect ratios \(c_i = n/m_i\) and scales \(s_i\) so each term has R-transform

\[R_i(w) = \lambda_i \frac{t_i}{(1 - t_i w},\]

with \(\lambda_i = \lambda w_i\). This is achieved by setting

\[c_i = 1 / \lambda_i, s_i = t_i \lambda_i.\]