Revisiting random tensor models at large N via the Schwinger-Dyson equations

TL;DR

This paper demonstrates that the Schwinger-Dyson equations uniquely determine the large N behavior of random tensor models, confirming their Gaussian nature with the full 2-point function as covariance.

Contribution

It proves that the algebra generated by SDEs in tensor models uniquely determines the physical solutions at large N, extending the Virasoro algebra concept from matrix models.

Findings

SDEs form a generalized algebra in tensor models

Linear SDEs determine a unique physical solution

Tensor models are Gaussian at large N

Abstract

The Schwinger-Dyson Equations (SDEs) of matrix models are known to form (half) a Virasoro algebra and have become a standard tool to solve matrix models. The algebra generated by SDEs in tensor models (for random tensors in a suitable ensemble) is a specific generalization of the Virasoro algebra and it is important to show that these new symmetries determine the physical solutions. We prove this result for random tensors at large N. Compared to matrix models, tensor models have more than a single invariant at each order in the tensor entries and the SDEs make them proliferate. However, the specific combinatorics of the dominant observables allows to restrict to linear SDEs and we show that they determine a unique physical perturbative solution. This gives a new proof that tensor models are Gaussian at large N, with the covariance being the full 2-point function.