Spectral asymptotics for contracted tensor ensembles

Let (Td,N) :Omega -> R-Nd be a random real symmetric Wigner-type tensor. For unit vectors (u(N)((i,j)) )(i is an element of I,j is an element of[d-2]) subset of SN-1, we study the contracted tensor ensemble (1/root N T-d,T-N [u(N)((i,1)) circle times center dot center dot center dot circle times u(N)((i,d-2))])(i is an element of I) For large N, we show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family (si)(i is an element of I) whose covariance (K-i;i'((N)))(i,i'is an element of I) is given by the rescaled overlaps of the corresponding symmetrized contractions K-i;i'((N)) = 1/d(d-1) < u(N)((i,1)) circle dot center dot center dot center dot circle dot u(N)((i,d) (2)), u(N)((i,1)) circle dot center dot center dot center dot circle dot u(N)((i,d) (2)) >, which is the true covariance of the ensemble up to a O-d(N-1) correction. We further characterize the extreme cases of the variance K-i,i((N)) is an element of [1/d!, 1/d(d-1)]. Our analysis relies on a tensorial extension of the usual graphical calculus for moment method calculations in random matrix theory, allowing us to access the independence in our random tensor ensemble.