  In this paper, we extend the recent analysis of the new large $D$ limit of
matrix models to the cases where the action contains arbitrary multi-trace
interaction terms as well as to arbitrary correlation functions. We discuss
both the cases of complex and Hermitian matrices, with
$\text{U}(N)^{2}\times\text{O}(D)$ and $\text{U}(N)\times\text{O}(D)$
symmetries respectively. In the latter case, the new large $D$ limit is
consistent for planar diagrams; at higher genera, it crucially requires the
tracelessness condition. For similar reasons, the large $N$ limit of tensor
models with reduced symmetries is typically inconsistent already at leading
order without the tracelessness condition. We also further discuss some
interesting properties of purely bosonic models pointed out recently and
explain that the standard argument predicting a non-trivial IR behaviour in
fermionic models \`a la SYK does not work for bosonic models. Finally, we
explain that the new large $D$ scaling is consistent with linearly realized
supersymmetry.
