Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (including the case where $X$ is an $N \times 1$ vector).

>**Question 1.** Is it true that for every $C>0$, there exists $c,A>0$ (only depending on $C$ and $\lambda_0$) such that
$P(s_\min(X) \le c\sqrt{N}) \le Ae^{-CN}$ ? What about the particular case when $C=1$ ?


In case the answer to the above is negative,

>**Question 2.** Find $A,c,C>0$ such that $P(s_\min(X) \le c\sqrt{N}) \le Ae^{-CN}$.

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Note that the case of Rademacher entries is solved in Theorem 2.7.1 [this document][1] (by T. Tao).


Update
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**Question 1** (and therefore, **Question 2**) has an affirmative answer with $A=2$ and without any constraint on $\lambda$ under than $\lambda < 1$. See https://mathoverflow.net/a/372119/78539


  [1]: https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf