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 $mathrm{U}(N)timesmathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $Ntimes N$ 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.