  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.
