Free cumulants and freeness for unitarily invariant random tensors

We address the question of the asymptotic description of random tensors that are local-unitary invariant, that is, invariant by conjugation by tensor products of independent unitary matrices. We consider both the mixed case of a tensor with $D$ inputs and $D$ outputs, and the case where there is a factorization between the inputs and outputs, called pure, which includes the random tensor models extensively studied in the physics literature. The finite size and asymptotic moments are defined using correlations of certain invariant polynomials encoded by $D$-tuples of permutations, up to relabeling equivalence. Finite size free cumulants associated to the expectations of these invariants are defined through invertible finite size moment-cumulants formulas. Two important cases are considered asymptotically: pure random tensors that scale like a complex Gaussian, and mixed random tensors that scale like a Wishart tensor. In both cases, we derive a notion of tensorial free cumulants associated to first order invariants, through moment-cumulant formulas involving summations over non-crossing permutations. The pure and mixed cases involve the same combinatorics, but differ by the invariants that define the distribution at first order. In both cases, the tensorial free-cumulants of a sum of two independent tensors are shown to be additive. A preliminary discussion of higher orders is provided. Tensor freeness is then defined as the vanishing of mixed first order tensorial free cumulants. The equivalent formulation at the level of asymptotic moments is derived in the pure and mixed cases, and we provide an algebraic construction of tensorial probability spaces, which generalize non-commutative probability spaces: random tensors converge in distribution to elements of these spaces, and tensor freeness of random variables corresponds to tensor freeness of the subspaces they generate.