Rank probabilities for real random N × N × 2 tensors

Abstract

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. © 2011 Association for Symbolic Logic.