The large $N$ factorization does not hold for arbitrary multi-trace observables in random tensors

TL;DR

This paper demonstrates that for random tensors of order three or higher, the large N factorization property of multi-trace observables fails generally, unlike in the matrix case, but holds for certain restricted families like melonic observables.

Contribution

It proves that large N factorization does not hold universally for multi-trace invariants in random tensors of order three or more, identifying conditions where it does.

Findings

Large N factorization fails for general multi-trace invariants in random tensors with D≥3.

Factorization holds for melonic observables, the dominant family at large N.

Contrasts the tensor case with the well-known matrix case where factorization always holds.

Abstract

We consider real tensors of order $D$, that is $D$-dimensional arrays of real numbers $T_{a^1a^2 \dots a^D}$, where each index $a^c$ can take $N$ values. The tensor entries $T_{a^1a^2 \dots a^D}$ have no symmetry properties under permutations of the indices. The invariant polynomials built out of the tensor entries are called trace invariants. We prove that for a Gaussian random tensor with $D\ge 3$ indices (that is such that the entries $T_{a^1a^2 \dots a^D}$ are independent identically distributed Gaussian random variables) the cumulant, or connected expectation, of a product of trace invariants is not always suppressed in scaling in $N$ with respect to the product of the expectations of the individual invariants. Said otherwise, not all the multi-trace expectations factor at large $N$ in terms of the single-trace ones and the Gaussian scaling is not subadditive on the connected components. This is in stark contrast to the $D=2$ case of random matrices in which the multi-trace expectations always factor at large $N$. The best one can do for $D\ge 3$ is to identify restricted families of invariants for which the large $N$ factorization holds and we check that this indeed happens when restricting to the family of melonic observables, the dominant family in the large $N$ limit.