Universality for Random Tensors

TL;DR

This paper establishes universality results for large random tensors, showing that their distribution converges to a Gaussian tensor model under broad conditions, extending known results from random matrices.

Contribution

It generalizes the universality principle from random matrices to tensors, including cases with invariant distributions and bounded cumulants.

Findings

Random tensors with i.i.d. entries converge to Gaussian tensor models.

Invariant distribution tensors also converge under bounded cumulants.

The covariance in the limit depends on the specific distribution details.

Abstract

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.