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

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 $\log P_N=(N^2/4)\log (e/4)+(\log N-1)/12-{\zeta }^{\prime }(-1)+\mathrm{O}(1/N)$ for large $N$, where $\zeta$ is the Riemann zeta function.

8 pages

References in corpus (1)

• Pfaffian point process for the Gaussian real generalised eigenvalue problem

Cited by in corpus (4)

• Probability of all eigenvalues real for products of standard Gaussian matrices
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• The average condition number of most tensor rank decomposition problems is infinite
• Real eigenvalue statistics for products of asymmetric real Gaussian matrices