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

TL;DR

This paper establishes a local limit law for the spectral distribution of the anticommutator of independent Wigner matrices, extending semicircle law techniques to a new matrix operation.

Contribution

It introduces a novel local limit law for the anticommutator of Wigner matrices, combining algebraic and analytical methods from free probability and random matrix theory.

Findings

Proves a local limit law similar to the semicircle law for the anticommutator.

Adapts techniques from Erdős-Yau-Yin for this new setting.

Provides a deterministic version of the local semicircle law.

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.