  Many-body localisation in interacting quantum systems can be cast as a
disordered hopping problem on the underlying Fock-space graph. A crucial
feature of the effective Fock-space disorder is that the Fock-space site
energies are strongly correlated -- maximally so for sites separated by a
finite distance on the graph. Motivated by this, and to understand the effect
of such correlations more fundamentally, we study Anderson localisation on
Cayley trees and random regular graphs, with maximally correlated disorder.
Since such correlations suppress short distance fluctuations in the disorder
potential, one might naively suppose they disfavour localisation. We find
however that there exists an Anderson transition, and indeed that localisation
is more robust in the sense that the critical disorder scales with graph
connectivity $K$ as $\sqrt{K}$, in marked contrast to $K\ln K$ in the
uncorrelated case. This scaling is argued to be intimately connected to the
stability of many-body localisation. Our analysis centres on an exact recursive
formulation for the local propagators as well as a self-consistent mean-field
theory; with results corroborated using exact diagonalisation.
