We prove two universality results for random tensors of arbitrary rank D. We
first prove that a random tensor whose entries are N^D independent, identically
distributed, complex random variables converges in distribution in the large N
limit to the same limit as the distributional limit of a Gaussian tensor model.
This generalizes the universality of random matrices to random tensors.
We then prove a second, stronger, universality result. Under the weaker
assumption that the joint probability distribution of tensor entries is
invariant, assuming that the cumulants of this invariant distribution are
uniformly bounded, we prove that in the large N limit the tensor again
converges in distribution to the distributional limit of a Gaussian tensor
model. We emphasize that the covariance of the large N Gaussian is not
universal, but depends strongly on the details of the joint distribution.Comment: Final versio
