# The $1/N$ expansion of tensor models with two symmetric tensors

## TL;DR

This paper develops a new method to establish a $1/N$ expansion for tensor models with two symmetric tensors, showing that melonic graphs dominate, even without the global jacket structure used in non-symmetric cases.

## Contribution

It introduces an approach to the $1/N$ expansion for symmetric tensor models that does not rely on jackets, proving melonic dominance for models with two symmetric tensors.

## Key findings

- The $1/N$ expansion is valid for symmetric tensor models.
- Melonic graphs dominate in the large N limit.
- The approach works for any tensor rank D.

## Abstract

It is well known that tensor models for a tensor with no symmetry admit a $1/N$ expansion dominated by melonic graphs. This result relies crucially on identifying \emph{jackets} which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the $1/N$ expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank $D$, a tensor model with two symmetric tensors and interactions the complete graph $K_{D+1}$ admits a $1/N$ expansion dominated by melonic graphs.