Local laws for polynomials of Wigner matrices

TL;DR

This paper establishes optimal local laws for eigenvalue distributions of self-adjoint polynomials in independent Wigner matrices, extending previous results to broader classes of polynomials and providing conditions that are numerically verifiable.

Contribution

It proves the bulk local law for general homogeneous degree-two polynomials and symmetrized products, generalizing previous results and offering practical conditions for broader polynomial classes.

Findings

Optimal local eigenvalue density matches global density for certain polynomials.

Established local laws for degree-two homogeneous polynomials and symmetrized products.

Provided numerically checkable conditions for more general polynomial ensembles.

Abstract

We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.