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

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.