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Universality for random tensors. (English. French summary) Zbl 1318.60010
Authors’ 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.

MSC:
60B99 Probability theory on algebraic and topological structures
60F99 Limit theorems in probability theory
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