Universality for random tensors

Gurau, Razvan

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1474-1525.

Nous démontrons deux théorèmes d’universalité pour les tenseurs aléatoires de rang $D$ quelconque. Nous montrons d’abord qu’un tenseur aléatoire dont les entrées sont $N^D$ variables complexes indépendantes identiquement distribuées converge en distribution dans la limite $N$ grand vers la même limite que la limite en distribution d’un modèle de tenseurs Gaussien. Cela généralise l’universalité des matrices aléatoires aux tenseurs aléatoires. Nous démontrons ensuite un deuxième théorème d’universalité, plus fort. Sous l’hypothèse que la distribution de probabilité jointe des entrées du tenseur est invariante, et en supposant que les cumulants de cette distribution invariante sont uniformément bornés, nous prouvons que dans la limite $N$ grand le tenseur converge à nouveau en distribution vers la même limite que la limite en distribution d’un modèle de tenseurs Gaussien. La covariance de la distribution Gaussienne à $N$ grand n'est pas universelle, mais dépend des détails de la distribution jointe.

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.

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