Top singular value of large random matrices: concentration results

Question

Let $A$ be a $n\times m$ random matrix, whose elements $a_{ij}$ are independent standard Gaussian random variables. I am interested in the case $n=\alpha N\,$, $\,m=(1-\alpha)N$ for $\alpha\in(0,1)$ fixed and $N\to\infty$.

Denote by $\sigma_\max(N)$ the largest singular value of $\frac{1}{\sqrt{N}}A$, that is the square root of the largest eigenvalue of the symmetric square matrix $\frac{1}{N}AA^T$.

I am looking for a concentration result of type: $$ \mathbb P(\sigma_\max(N)\geq K) \,\leq\, e^{-N F(K)}$$ for constants $K$ in a suitable range, a suitable function $F(K)$, and $N$ sufficiently large. I do not need $F$ to be optimal.

In the square matrix case ($\alpha=\frac{1}{2}$), an analogous result holds true for the top eigenvalue $\lambda_\max(N)$ of the symmetric matrix $\frac{A+A^T}{\sqrt{2N}}$. But I could not find anything about large deviations of the top singular value in the rectangular case. Any reference or suggestion is very welcome.

Answer 1

This largest singular value is the norm of the matrix. You can use a net argument to show that there is a $C$ so that $$\mathbb{P}(\| A \|_{op} \geq C\sqrt{N}(\sqrt{\alpha} + \sqrt{1 - \alpha} + t)) \leq 2 e^{-Nt^2}$$ for all $\alpha, t$. For a reference, this appears as Theorem 4.4.5 in Vershynin's High Dimensional Probability book.