A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices

Abstract

Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from one of the recent papers of Erd\"os-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, a self-adjointness-preserving variant of the linearization trick due to Haagerup-Schultz-Thorbj\o rnsen, and the Schwinger-Dyson equation. A byproduct of our work is a relatively simple deterministic version of the local semicircle law.