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

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Rank probabilities for real random $N\times N \times 2$ tensors
2.	Creator	Author's name, affiliation, country	Göran Bergqvist; Linköping University
2.	Creator	Author's name, affiliation, country	Peter J. Forrester; University of Melbourne
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	tensors; multi-way arrays; typical rank; random matrices
3.	Subject	Subject classification	15A69; 15B52; 60B20
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	the Australian Research Council
7.	Date	(YYYY-MM-DD)	2011-10-21
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1655
10.	Identifier	Digital Object Identifier	10.1214/ECP.v16-1655
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 16
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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