Tensor models from the viewpoint of matrix models: the cases of loop models on random surfaces and of the Gaussian distribution

Abstract

Two direct connections between random tensors and random matrices are discussed in this article. In the rst part, we introduce U($\tau$) matrix models which generate fully packed, oriented loops on random surfaces. e latter are found to be in bijection with a set of regular edge-colored graphs. 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 applied in this context. For example, congurations which maximize the number of loops are precisely the melonic graphs of tensor models and a scaling limit which projects onto themelonic sector is found. is approach is generalized to higher-rank tensor models, which generate loops with fugacity $\tau$ on triangulations in dimension $d–1$. In the second part, we introduce singular value decompositions to evaluate the expectations of polynomial observables of Gaussian random tensors. Performing the integrals over the unitary group leads to a notion of eective observables which expand onto regular trace invariants. We show that both asymptotic and exact new calculations of expectations can be performed this way.