Local semicircle law for Curie-Weiss type ensembles

Abstract

We derive local semicircle laws for random matrices with exchangeable entries which exhibit correlations that decay slowly in the dimension N of the matrix. To be precise, any ℓ-point correlation 𝔼[Y1⋯Yℓ] between distinct matrix entries Y1,…,Yℓ may decay at a rate of only N−ℓ∕2. We call our ensembles of (high temperature) Curie-Weiss type, and Curie-Weiss(β)-distributed entries directly fit within our framework in the high temperature regime β∈[0,1]. Using rank-one perturbations, we show that even in the low-temperature regime β∈(1,∞), where ℓ-point correlations survive in the limit, the local semicircle law still holds after rescaling the matrix entries with a constant which depends on β but not on N.