Isotropic self-consistent equations for mean-field random matrices

We present a simple and versatile method for deriving (an)isotropic local laws for general random matrices constructed from independent random variables. Our method is applicable to mean-field random matrices, where all independent variables have comparable variances. It is entirely insensitive to the expectation of the matrix. In this paper we focus on the probabilistic part of the proof - the derivation of the self-consistent equations. As a concrete application, we settle in complete generality the local law for Wigner matrices with arbitrary expectation.