Spectra of contractions of the Gaussian Orthogonal Tensor Ensemble

Abstract

In this article, we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of order $r$ and dimension $n$ drawn from the density \[ f(\mathcal{G}) \propto \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg). \] For $\mathbf{w}\in {\mathbb{S}}^{n-1}$, the unit-sphere in ${\mathbb{R}}^n$, we consider the matrix-valued contraction $\mathcal{G}\cdot {\mathbf{w}}^{\otimes (r-2)}$ when both $r$ and $n$ go to infinity such that $r/n\to c\in [0,\infty ]$. We obtain semi-circle bulk-limits in all regimes, generalising the works of Goulart et al. (2022); Au and Garza-Vargas (2023); Bonnin (2024) in the fixed-$r$ setting. We also study the edge-spectrum. We obtain a Baik-Ben Arous-Péché phase-transition for the largest and the smallest eigenvalues at $r=4$, generalising a result of Mukherjee et al. (2024) in the context of adjacency matrices of random hypergraphs. We also show that the extreme eigenvectors of $\mathcal{G}\cdot {\mathbf{w}}^{\otimes (r-2)}$ contain non-trivial information about the contraction direction $\mathbf{w}$. Finally, we report some results, in the case $r=4$, on mixed contractions $\mathcal{G}\cdot \mathbf{u}\otimes \mathbf{v}$, $\mathbf{u},\mathbf{v}\in {\mathbb{S}}^{n-1}$. While the total variation distance between the joint distribution of the entries of $\mathcal{G}\cdot \mathbf{u}\otimes \mathbf{v}$ and that of $\mathcal{G}\cdot \mathbf{u}\otimes \mathbf{u}$ goes to $0$ when $\Vert \mathbf{u}-\mathbf{v}\Vert =o(n^{-1})$, the bulk and the largest eigenvalues of these two matrices have the same limit profile as long as $\Vert \mathbf{u}-\mathbf{v}\Vert =o(1)$. Furthermore, it turns out that there are no outlier eigenvalues in the spectrum of $\mathcal{G}\cdot \mathbf{u}\otimes \mathbf{v}$ when $⟨\mathbf{u},\mathbf{v}⟩=o(1)$.

45 pages, 1 figure; abstract shortened to meet arXiv requirements