R-transform associated with asymptotic negative spectral moments of Jacobi ensemble

Abstract

We derive an explicit formula for the R-transform of inverse Jacobi matrix I + W₁^-1 W₂, where W₁,W₂ ~ W_p(I, n_i), i = 1, 2 are independent and I is pxp dimensional identity matrix using property of asymptotic freeness of Wishart and deterministic matrices. Procedure can be extended to other sets of the asymptotically free independent matrices. Calculations are illustrated with some simulations on fixed size matrices.

Keywords: Jacobi ensemble; R-transform; S-transform; Negative spectral moments; Spectral moments; Wishart matrix; Marčenko–Pastur law; asymptotical freeness