Title: Eigenvalue approximation of sums of Hermitian matrices from eigenvector localization/delocalization,
Abstract: We propose a technique for calculating and understanding the eigenvalue
distribution of sums of random matrices from the known distribution of the
summands. The exact problem is formidably hard. One extreme approximation to
the true density amounts to classical probability, in which the matrices are
assumed to commute; the other extreme is related to free probability, in which
the eigenvectors are assumed to be in generic positions and sufficiently large.
In practice, free probability theory can give a good approximation of the
density.
We develop a technique based on eigenvector localization/delocalization that
works very well for important problems of interest where free probability is
not sufficient, but certain uniformity properties apply. The
localization/delocalization property appears in a convex combination parameter
that notably, is independent of any eigenvalue properties and yields accurate
eigenvalue density approximations.
We demonstrate this technique on a number of examples as well as discuss a
more general technique when the uniformity properties fail to apply.