Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries. Let $\sigma_1(A)$ be the largest singular value of $A$ and let $t \ge 0$.

Question. What is the the value (of good upper bound) of $\mathbb E_A[e^{-t\sigma_1(A)}]$ ?

Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral norm of $A$) and let $t \ge 0$.

Question. What is a good upper bound for $\mathbb E_A[e^{-t\|A\|_2}]$ ?