RANK PROBABILITIES FOR REAL RANDOM N x N x 2 TENSORS
نویسندگان
چکیده
We prove that the probability PN for a real random Gaussian N ×N ×2 tensor to be of real rank N is PN = (Γ((N + 1)/2))N/G(N + 1), where Γ(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 πN/2 for N even. The probability to be of rank N + 1 is 1− PN . The proof makes use of recent results on the probability of having k real generalized eigenvalues for real random Gaussian N×N matrices. We also prove that log PN = (N2/4) log(e/4) + (log N − 1)/12− ζ′(−1) +O(1/N) for large N , where ζ is the Riemann zeta function.
منابع مشابه
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