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 $t \ge 0$.



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> **Question.** What is a good upper bound for $\mathbb E\_A[e^{-t\|A\|\_2}]$ ?
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Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$
The probability distribution of the largest singular value of $A$ (or the largest eigenvalue $\lambda\_1$ of $AA^T$) is derived in [Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution](https://arxiv.org/abs/1209.3394). There are exact answers for finite $n,m$ and asymptotic forms for large $n,m$. From here finding the desired average of $e^{-t\lambda\_1}$ is a matter of quadrature.


