Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be the rank of $\mathbf{R}$. Three questions:

(1) can we say $r = \min(m,n)$?

(2) what is the probability of having a unitary submatrix of size $r$ inside $\mathrm{R}$?

(3) if the answer of question 2 is not available in general, is there any asymptotic result? For instance, existence of at least one unitary submatrix of size $r$ almost surely as $n\to \infty$?

PS: what about ``almost unitary'' submatrices for questions (2) and (3)?A definition for almost unitary matrix can be that, $\mathbf{U}^+ = \mathbf{U}^H + \mathbf{e}$, where $\mathbf{e}$ is a small error matrix (small entries), and $+$ is the Moore–Penrose pseudoinverse operation.