Does free probability have anything to say about the eigenvalue correlations of random matrices?

Question

Free probability provides a compact route to compute the average eigenvalue density for various families of random matrices in the large $N$ limit. Does it provide any route to eigenvalue correlations, which are generally useful in the same limit?

Answer 1

Eigenvalue correlations on the macroscopic level (fluctuations of linear statistics) are in free probability theory captured by the notion of "second order freeness"; a treatment of this can be found in chapter 5 of the book Mingo, Speicher: Free Probability and Random Matrices

Answer 2

The first reference is a Ph.D. thesis, probably the best entry point to the literature of free probability approaches to eigenvalue/eigenvector correlations; the other references involve particular applications.

• Free probability approach to microscopic statistics of random matrix eigenvalues (2017)

• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem (2016)

• Complete diagrammatics of the single ring theorem (2017)