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 $𝔼[Y_1\cdots Y_{\ell }]$ between distinct matrix entries $Y_1,\dots ,Y_{\ell }$ may decay at a rate of only $N^{-\ell ∕2}$. 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 $\mathit{\beta }\in [0,1]$. Using rank-one perturbations, we show that even in the low-temperature regime $\mathit{\beta }\in (1,\mathrm{\infty })$, 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.