I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting distribution.
Random symmetric tensor
Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\in \mathbb{R}$ for all $1\leq i,j,k\leq N$. By symmetric I mean that if I permute any indices the value stays the same: \begin{equation} T_{ijk}=T_{ikj}=\dots=T_{jki} \end{equation}
Eigenvalues
One possible definition of eigenvalues: let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that: \begin{equation} \sum_{jk}^NT_{ijk}x_kx_j=\lambda x_i \label{eq1} \end{equation} with the constraint that $\sum_i x_i^2=1$.
It has been shown (in [1] for example) that for a tensor of order 3 and size $N$ there are $2^{N}-1$ eigenvalues $\lambda$. Perhaps an intuitive idea on why is to see that the eigenvalue equation is composed of $N$ quadratic equations. Each equation possesses two roots, hence $2^N$ possible combinations of solutions, minus the normalisation constraint.
Possible leads
It would be interesting to find the limiting distribution of eigenvalues of random tensors for large $N$. The eigenvalues are real only if their eigenvectors are real. I conjecture that the purely real eigenvalues scale with $\sqrt{N}$ whereas the general eigenvalues scale with $N$. That is to say, I would expect the real eigenvalues to be distributed on a closed non trivial interval when the elements of the tensor have a variance scaled with $1/\sqrt{N}$.
1. Reducing it to a non-linear matrix problem:
We define the matrix $M$ such that: $M_{ij}=\sum_{k}T_{ijk}x_k$. Therefore the eigenvalue problem would boil down to find the eigenvalues of $M$:
\begin{align} \sum_{jk}^NT_{ijk}x_kx_j&=\lambda x_i\\ \implies \sum_{j}M_{ij}x_j&=\lambda x_i \end{align}
Where the elements of $M$ now also depend on the eigenvectors.
2. Choosing an appropriate basis to uncouple the equations:
If we fix the index $i$, we can represent $T$ as a collection of $N$ matrices: ${M}^{(1)},{M}^{(2)}\dots{M}^{(N)}$ where $T_{ijk}={M_{jk}}^{(i)}$.
Since $T$ is real and fully symmetric, ${M}^{(i)}$ is also real symmetric $(1\leq i\leq N)$. There will also share some elements in common and not be fully independent from one another. For example $T_{ijk}=T_{kij}\implies M_{ij}^{(1)}=M_{1j}^{(i)}$. So matrix $M^{(1)}$ will be composed of the first column of all the other matrices. What is the relation between the eigenvalues of $M^{(i)}$ and $M^{(j)}$? Are they directly linked to the eigenvalues of $T$?
Here were some of the first ideas that came naturally to my mind. If you know a paper or some ideas on how to tackle this problem this would be very interesting.
[1]: Cartwright, D., & Sturmfels, B. (2013). The number of eigenvalues of a tensor. Linear algebra and its applications, 438(2), 942-952.
