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 $1/\sqrt{D}$ in $\mathrm{U}(N)\times\mathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $N\times N$ matrices transform in the vector representation of $\mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $\eta(h)$ on the maximum power $D^{1+\eta(h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ 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 $\eta(h)\leq 2h$; 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.