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Freeness for tensors
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We pursue the current developments in random tensor theory by laying the foundations of a free probability theory for tensors and establish its relevance in the study of random tensors of high dimension. We give a definition of freeness associated to a collection of tensors of possibly different orders. Our definition reduces to the usual freeness when only tensors of order 2 are concerned. We define the free cumulants which are associated to this notion of tensor freeness. We prove that the basic models of random tensors are asymptotically free as the dimension goes to infinity. On the way, we establish Schwinger-Dyson loop equations associated to random tensors.
Forward citations
Cited by 2 Pith papers
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The large $N$ factorization does not hold for arbitrary multi-trace observables in random tensors
Gaussian random tensors of order D >= 3 admit multi-trace observables whose joint cumulants scale in N as strongly as the product of their one-trace expectations, so large N factorization fails in general.
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Additional constraints for the tensor bootstrap
New positivity constraints from open bubbles and color matrices provide sharp bounds on unitary tensor integrals at finite N and probe deviations from Gaussian universality.
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