New 1/N expansions in random tensor models

Razvan Gurău

Razvan Gurău

Razvan Gurău

Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.

Valentin Bonzom, Frédéric Combes

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.

Wei Li, Bin-Bin Chen, yun-jing liu +1 more

Razvan Gurău

Matrix models are a highly successful framework for the analytic study of random two dimensional surfaces with applications to quantum gravity in two dimensions, string theory, conformal field theory, statistical physics in random geometry, etc. Their success relies crucially on the so called 1/N expansion introduced by 't Hooft. In higher dimensions matrix models generalize to tensor models. In the absence of a viable 1/N expansion tensor models have for a long time been less successful in providing an analytically controlled theory of random higher dimensional topological spaces. This situation has drastically changed recently. Models for a generic complex tensor have been shown to admit a 1/N expansion dominated by graphs of spherical topology in arbitrary dimensions and to undergo a phase transition to a continuum theory.