Fluctuations of functions of Wigner matrices

Summary

We prove a CLT for the entrywise fluctuations of functions of random matrices. The use of Pleijel's formula makes our result applicable to all functions of bounded variation.

Abstract

We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation. Our result holds for any function $f$ of the matrix that has bounded variation and thus considerably relaxes the regularity requirement imposed in [7,11].

Paper

1610.07084.pdf