On the large D expansion of Hermitian multi-matrix models

We investigate the existence and properties of a double asymptotic expansion in 1/N-2 and 1/root D in U(N) x O(D) invariant Hermitian multi-matrix models, where the N x N matrices transform in the vector representation of O(D). The crucial point is to prove the existence of an upper bound eta(h) on the maximum power D1+eta(h) of D that can appear for the contribution at a given order N2-2h in the large N expansion. We conjecture that eta(h) = h in a large class of models. In the case of traceless Hermitian matrices with the quartic tetrahedral interaction, we are able to prove that eta(h) <= 2h; the sharper bound eta(h) = h is proven for a complex bipartite version of the model, with no need to impose a tracelessness condition. We also prove that eta(h) = h for the Hermitian model with the sextic wheel interaction, again with no need to impose a tracelessness condition.