Tensor models from the viewpoint of matrix models: the case of the Gaussian distribution

TL;DR

This paper introduces a method using singular value decompositions to evaluate expectations of polynomial observables in Gaussian random tensors, simplifying complex calculations in tensor models.

Contribution

It presents a novel approach leveraging matrix model techniques to compute expectations in Gaussian tensor models, bridging tensor and matrix model frameworks.

Findings

Effective observables expand onto matrix trace invariants.

Asymptotic and exact expectation calculations are demonstrated.

Method simplifies evaluation of polynomial observables in Gaussian tensors.

Abstract

Observables in random tensor theory are polynomials in the entries of a tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution. In this article, we introduce singular value decompositions to evaluate the expectations of polynomial observables of Gaussian random tensors. Performing the matrix integrals over the unitary group leads to a notion of effective observables which expand onto regular, matrix trace invariants. Examples are given to illustrate that both asymptotic and exact new calculations of expectations can be performed this way.