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

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 recent papers of Erdös-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, the linearization trick due to Haagerup-Schultz-Thorbjørnsen in a self-adjointness-preserving variant and the Schwinger-Dyson equation. A by-product of our work is a relatively simple deterministic version of the local semicircle law.