Low order maximally single-trace graphs as the first counterexamples to large N factorization in random tensors

TL;DR

This paper presents the first low-order examples of specific 3-regular, 3-edge-colored graphs that serve as counterexamples to the large N factorization property in Gaussian random tensor models, highlighting a fundamental difference from matrix models.

Contribution

It introduces the first explicit low-order graphs demonstrating non-factorization in tensor models, providing concrete counterexamples to the large N factorization conjecture.

Findings

First explicit counterexamples to large N factorization in tensors

Low order 3-regular 3-edge-colored graphs demonstrate non-factorization

Contrasts tensor model behavior with matrix models

Abstract

We give the first and lowest order examples of 3-regular 3-edge-colored graphs that demonstrate the non-factorization of tensor model invariants in the large N limit of Gaussian random tensors, as proven on general grounds in [Gurau R., Joos F. and Sudakov B., Lett. Math. Phys., 115 (2025), arXiv:2506.15362 [math-ph]]. This non-factorization is in stark contrast to the well-known large N factorization for random matrices.