Splitting-merging transitions in tensor-vectors systems in exact large-$N$ limits

Matrix models have phase transitions in which distributions of variables change topologically like the Gross-Witten-Wadia transition. In a recent study, similar splitting-merging behavior of distributions of dynamical variables was observed in a tensor-vectors system by numerical simulations. In this paper, we study the system exactly in some large-$N$ limits, in which the distributions are discrete sets of configurations rather than continuous. We find cascades of first-order phase transitions for fixed tensors, and first- and second-order phase transitions for random tensors, being characterized by breaking patterns of replica symmetries. The system is of interest across three different subjects at least: The splitting dynamics plays essential roles in emergence of classical spacetimes in a tensor model of quantum gravity; The splitting dynamics automatically detects the rank of a tensor in the tensor rank decomposition in data analysis; The system provides a variant of the spherical $p$-spin model for spin glasses with a new non-trivial parameter. We discuss some implications of the results from these perspectives. The results are compared with some numerical simulations to check the large-$N$ convergence and the assumptions made in the analysis.