My understanding is that Anderson localization is highly generic for non-interacting Fermi systems in low dimension. More precisely, let us consider the Anderson model for fermions, with Hamiltonian
$$H=\sum_{n}\epsilon_{n}|n\rangle\langle n| +t\sum_{\langle ij\rangle}|i\rangle\langle j|$$
where $\langle ij\rangle$ denotes nearest neighbors on some lattice (we can take this to be a $d$-dimensional cubic lattice) and $\epsilon_{n}$ are random variables describing the on-site energy. When $\epsilon_{n}$ is drawn from a uniform distribution, there is a substantial body of work indicating that the conductivity of this model vanishes for $d\leq 2$ - that is, all of the Hamiltonian eigenstates are localized, with wavefunctions that decay exponentially as a function of distance.
My question is if localization holds when the $\epsilon_{n}$ are correlated. More precisely, if we define a probability distribution $P(\epsilon_{1},\cdots,\epsilon_{L})$ for the on-site energies, is it believed/known/proven that localization holds in low dimensions for any choice of $P$? At the end of the day I'd be happy to know the answer to this question at a physicist's level of rigor, but it would also be nice to get a sense of what is rigorously known in the theory of localization more generally.
Thanks in advance.
