Spectral asymptotics for contracted tensor ensembles

Abstract

Let ${\mathbf{T}}_{d,N}:\mathrm{\Omega }\to {\mathbb{R}}^{N^d}$ be a random real symmetric Wigner-type tensor. For unit vectors ${(u_N^{(i,j)})}_{i\in I,j\in [d-2]}\subset {\mathbb{S}}^{N-1}$, we study the contracted tensor ensemble

$${\left(\frac{1}{\sqrt{N}}{\mathbf{T}}_{d,N}\left[u_N^{(i,1)}\otimes \cdots \otimes u_N^{(i,d-2)}\right]\right)}_{i\in I}.$$

For large N, we show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family ${(s_i)}_{i\in I}$ whose covariance ${({\mathbf{K}}_{i,i^{\prime }}^{(N)})}_{i,i^{\prime }\in I}$ is given by the rescaled overlaps of the corresponding symmetrized contractions

$${\mathbf{K}}_{i,i^{\prime }}^{(N)}=\frac{1}{d(d-1)}⟨u_N^{(i,1)}\odot \cdots \odot u_N^{(i,d-2)},u_N^{(i^{\prime },1)}\odot \cdots \odot u_N^{(i^{\prime },d-2)}⟩,$$

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 ${\mathbf{K}}_{i,i}^{(N)}\in [\frac{1}{d!},\frac{1}{d(d-1)}]$. 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.