Tensor models from the viewpoint of matrix models: the case of loop models on random surfaces

Abstract

We study a connection between random tensors and random matrices through $U(\tau)$ matrix models which generate fully packed, oriented loops on random surfaces. The latter are found to be in bijection with a set of regular edge-colored graphs typically found in tensor models. It is shown that the expansion in the number of loops is organized like the 1/N expansion of rank-three tensor models. Recent results on tensor models are reviewed and applied in this context. For example, configurations which maximize the number of loops are precisely the melonic graphs of tensor models and a scaling limit which projects onto the melonic sector is found. We also reinterpret the double scaling limit of tensor models from the point of view of loops on random surfaces. This approach is eventually generalized to higher-rank tensor models, which generate loops with fugacity $\tau$ on triangulations in dimension $d-1$.