Spectral asymptotics for contracted tensor ensembles

Let (formula presented) be a random real symmetric Wigner-type tensor. For unit vectors (formula presented), we study the contracted tensor ensemble (formula presented) For large N, we show that the joint spectral distribution of this ensemble is wellapproximated by a semicircular family (formula presented) whose covariance (formula presented) is given by the rescaled overlaps of the corresponding symmetrized contractions (formula presented), which is the true covariance of the ensemble up to (formula presented) correction. We further characterize the extreme cases of the variance (formula presented) 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.