Rank probabilities for real random $N\times N\times 2$ tensors

TL;DR

This paper derives an exact formula for the probability that a real Gaussian tensor of size N×N×2 has a specific rank, connecting tensor rank probabilities with random matrix eigenvalue distributions.

Contribution

It provides a closed-form expression for the probability of a real Gaussian tensor having rank N, linking tensor rank to eigenvalue statistics of random matrices.

Findings

Exact probability formula involving gamma and Barnes G-functions.

Asymptotic expression for large N involving the Riemann zeta function.

Probability of rank N+1 is complementary to rank N.

Abstract

We prove that the probability $P_N$ for a real random Gaussian $N\times N\times 2$ tensor to be of real rank $N$ is $P_N=(\Gamma((N+1)/2))^N/G(N+1)$, where $\Gamma(x)$, $G(x)$ denote the gamma and Barnes $G$-functions respectively. This is a rational number for $N$ odd and a rational number multiplied by $\pi^{N/2}$ for $N$ even. The probability to be of rank $N+1$ is $1-P_N$. The proof makes use of recent results on the probability of having $k$ real generalized eigenvalues for real random Gaussian $N\times N$ matrices. We also prove that $\log P_N= (N^2/4)\log (e/4)+(\log N-1)/12-\zeta '(-1)+{\rm O}(1/N)$ for large $N$, where $\zeta$ is the Riemann zeta function.