Here is one reference: Free probability approach to microscopic statistics of random matrix eigenvalues, a Ph.D. thesis which you could use as an entry point to the literature (which is extensive).

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)