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2011 Rank probabilities for real random $N\times N \times 2$ tensors
Göran Bergqvist, Peter Forrester
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Electron. Commun. Probab. 16: 630-637 (2011). DOI: 10.1214/ECP.v16-1655

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.

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Göran Bergqvist. Peter Forrester. "Rank probabilities for real random $N\times N \times 2$ tensors." Electron. Commun. Probab. 16 630 - 637, 2011. https://doi.org/10.1214/ECP.v16-1655

Information

Accepted: 21 October 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1245.15036
MathSciNet: MR2846656
Digital Object Identifier: 10.1214/ECP.v16-1655

Subjects:
Primary: 15A69
Secondary: 15B52 , 60B20

Keywords: multi-way arrays , random matrices , tensors , typical rank

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