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

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