Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

We study a matrix model that has $\phi_a^i\ (a=1,2,\ldots,N,\ i=1,2,\ldots,R)$ as its dynamical variable, whose lower indices are pairwise contracted, but upper ones are not always done so. This matrix model has a motivation from a tensor model for quantum gravity, and is also related to the physics of glasses, because it has the same form as what appears in the replica trick of the spherical $p$-spin model for spin glasses, though the parameter range of our interest is different. To study the dynamics, which in general depends on $N$ and $R$, we perform Monte Carlo simulations and compare with some analytical computations in the leading and the next-leading orders. A transition region has been found around $R\sim N^2/2$, which matches a relation required by the consistency of the tensor model. The simulation and the analytical computations agree well outside the transition region, but not in this region, implying that some relevant configurations are not properly included by the analytical computations. With a motivation coming from the tensor model, we also study the persistent homology of the configurations generated in the simulations, and have observed its gradual change from $S^1$ to higher dimensional cycles with the increase of $R$ around the transition region.