Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rank probabilities for real random $N\times N\times 2$ tensors   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.

13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs