Universality for Random Tensors

Details Universality for Random Tensors Universality for Random Tensors

2011-11-02

arxiv.org/abs/1111.0519v2

Mathematics

Probability

Second version. The paper has been extensively rewritten (and an appendix added) to improve readability. Results slightly gene

Scientific paper

We prove two universality results for random tensors of arbitrary rank D. We first prove that, assuming that the tensor entries are N^D independent identically distributed complex random variables then in the large N limit we obtain a tensor distributed on a Gaussian. 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, we prove that in the large N limit we obtain again a tensor distributed on a Gaussian. We emphasize that the covariance of the large N Gaussian is not universal, but depends strongly on the details of the joint distribution.

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