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

Bonzom, Valentin; Combes, Frédéric

Annales de l’Institut Henri Poincaré D, Tome 2 (2015) no. 1, pp. 1-47.

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, con�gurations 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 e�ective observables which expand onto regular trace invariants. We show that both asymptotic and exact new calculations of expectations can be performed this way.

Classification : 05-XX, 60-XX
Mots-clés : Random matrices, fully packed loop models, random tensors, 1/$N$ expansion