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.
 
 - <A HREF="http://kups.ub.uni-koeln.de/7662/">Free probability approach
   to microscopic statistics of random matrix eigenvalues</A> (2017)
   
 - <A HREF="https://arxiv.org/abs/1608.04923">Squared eigenvalue
   condition numbers and eigenvector correlations from the single ring
   theorem</A> (2016)

 - <A HREF="https://arxiv.org/abs/1704.07719">Complete diagrammatics of the single ring theorem</A> (2017)