Spectral asymptotics for contracted tensor ensembles

Abstract

Let ${\mathbf{T}}_{d,N}:\Omega \to {\mathbb{R}}^{N^d}$ 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 \[ \mathbf{K}_{i, i'}^{(N)} = \frac{1}{d(d-1)}\langle u_N^{(i, 1)} \odot \cdots \odot u_N^{(i, d-2)}, u_N^{(i', 1)} \odot \cdots \odot u_N^{(i', d-2)} \rangle, \] 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.

v3: updated to incorporate feedback from referees, including a shorter proof of Proposition 4.2; 31 pages, 7 figures

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