Consider a tensor product
$ V^{\otimes n} = \underbrace{V\otimes\cdots\otimes V}_{n} $
where $V$ is a vector space over $\mathbb R$, $\dim V = m$ , hence $\dim V^{\otimes n} = m^n$ .
So every $A \in V^{\otimes n}$ can be represented as
$A = \sum_{i=1}^r a^i_1 \otimes a^i_2 \ldots \otimes a^i_n, \;\;\; a_i \in V $
in a non-unique way. Taking $R$ to be minimum $r$ among all the possible decompositions of A.
$R = \min \left \{ r : A = \sum_{i=1}^r a^i_1 \otimes a^i_2 \ldots \otimes a^i_n, \;\;\; a_i \in V \right \}$
How many tensors have certain $R$ ? How many tensors have $R=1$? Or $R = m^n$ ? What is the typical $R$ (mean, median mean, the most probable), what is the distribution?
IMPORTANT How should I imagine (picture) tensors for which $R$ is (near) maximum? What hinders them from decomposition?
Maybe there are some experimental data. I'm mostly interested in high $m$'s and $n$'s, though every answer is welcome.