Typical ranks of random order-three tensors

In this paper we study typical ranks of real $m\times n\times \mathrm{\ell }$ tensors. In the case $(m-1)(n-1)+1\le \mathrm{\ell }\le mn$ the typical ranks are contained in $\{\mathrm{\ell },\mathrm{\ell }+1\}$, and $\mathrm{\ell }$ is always a typical rank. We provide a geometric proof of this fact. We express the probabilities of these ranks in terms of the probabilities of the numbers of intersection points of a random linear space with the Segre variety. In addition, we give some heuristics to understand how the probabilities of these ranks behave, based on asymptotic results on the average number of real points in a random linear slice of a Segre variety with a subspace of complementary dimension. The typical ranks of real $3\times 3\times 5$ tensors are $5$ and $6$. We link the rank probabilities of a $3\times 3\times 5$ tensor with i.i.d.\ Gaussian entries to the probability of a random cubic surface in ${\mathrm{\P }}^3$ having real lines. As a consequence, we get a bound on the expected number of real lines on such a surface.