Universality for Random Tensors

By R. Gurau. 2011

136 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Radiology, Nuclear Medicine and Imaging, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4725dd6f9c0cdfea614ba751b0580f74307e0131

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In Computational Mathematics

In Radiology, Nuclear Medicine and Imaging

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Citations in the corpus

A Renormalizable 4-Dimensional Tensor Field Theory (2011, background): “This theory and this symmetry has been proved universal in the precise sense of probability theory: every independentidentically-distributed or even invariant probability law on uncolored random tensors is governed in the large N limit by the 1/N expansion of colored models [50].”
Random tensor models in the large N limit: Uncoloring the colored tensor models (2012, background): “The algebra of these operators has been discussed at length in [56].”
O(N) Random Tensor Models (2015, background): “Many interesting results have been gathered for both colored or uncolored tensor models, including the characterization of next-to-leading order Feynman graphs in the large N expansion and the definition of a double-scaling limit [10–12], or universality results [13] proven with constructive methods”