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

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.

Figures (4)

• Figure 1: Illustration of the face structure considered in the proof of Lemma \ref{['lem:flip']}. See there for further explanation. The dotted lines indicate how the edges are connected along the rest of the faces $\mathcal{C}$ (of colors $(1,2)$) and $\mathcal{C}'$ (of colors $(1,3)$).
• Figure 2: Illustration of the construction of the matching $M'$ in the proof of Lemma \ref{['lem:flip']}. See there for further explanations. We only show the color $0$ edges that are different in $M$ (in blue) and $M'$ (in green). All other color $0$ edges are parallel to the color $1$ edges. As before, the dotted lines indicate how the edges are connected along the faces.
• Figure 3: All 3-regular 3-edge-colored graphs on $2n=16$ vertices with $\max_M F =12$.
• Figure 4: Continuation of Fig. \ref{['fig:graphs1']}.

Theorems & Definitions (12)

• Definition
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof