Seminar on Free Probability
and Random Matrices

Winter 2022

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 $(\mathbf{K}_{i, i'}^{(N)})_{i, i'\in I}$ 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. In particular, if the limits $\mathbf{K}_{i, i'} = \lim_{N\to \infty} \mathbf{K}^{(N)}_{i, i'}$ exist, then the contracted tensor ensemble converges in distribution almost surely to a semicircular family $(s_i)_{i\in I}$ of covariance $(\mathbf{K}_{i, i'})_{i, i'\in I}$. In the single-matrix model $\#(I) = 1$, this implies that the empirical spectral distribution of the contracted tensor ensemble is close to the semicircle distribution \[\frac{1}{2\pi \mathbf{K}_{i, i}^{(N)}} (4\mathbf{K}_{i, i}^{(N)} - x^2)_+^{1/2}\, dx\] in Kolmogorov-Smirnov distance with high probability. 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. This is joint work with Jorge Garza-Vargas.