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Rank probabilities for real random NxNx2 tensors
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
University of Melbourne, Victoria, Australia.
2011 (English)In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 16, 630-637 p.Article in journal (Refereed) Published
Abstract [en]

We prove that the probability P_N for a real random Gaussian NxNx2 tensor to be of real rank N is P_N=(Gamma((N+1)/2))^N/G(N+1), where Gamma(x) and G(x) denote the gamma and the 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 x N matrices. We also prove that log P_N= (N^2/4)log (e/4)+(log N-1)/12-zeta'(-1)+O(1/N) for large N, where zeta is the Riemann zeta function.

Place, publisher, year, edition, pages
Institute of Mathematical Statistics / Bernoulli society. , 2011. Vol. 16, 630-637 p.
Keyword [en]
tensors, multi-way arrays, typical rank, random matrices
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-72024DOI: 10.1214/ECP.v16-1655ISI: 000296162600001OAI: oai:DiVA.org:liu-72024DiVA: diva2:456014
Note

Funding Agencies|Australian Research Council||

Available from: 2011-11-11 Created: 2011-11-11 Last updated: 2017-12-08

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Bergqvist, Göran

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CiteExportLink to record
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Citation style
  • apa
  • harvard1
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  • vancouver
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  • Other style
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Language
  • de-DE
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Output format
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