# Top singular value of large random matrices: concentration results

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.

• Does the Marchenko-Pastur distribution help? Nov 8, 2021 at 8:41
• This is not exactly what you're looking for, but chapter 5.2 in 'Spectral Analysis of Large Dimensional Random Matrices' by Bai and Silverstein gives an almost sure limit of the largest singular value (which is the larges eigenvalue of an empirical covariance matrix). Nov 8, 2021 at 10:11
• PS. Would you mind giving me a reference for the result for square matrices? Nov 8, 2021 at 10:16
• @Tardis for $n=m$ the rate function for large deviations of the top eigenvalue is computed here link.springer.com/content/pdf/10.1007/PL00008774.pdf, Theorem 6.2 . But if you do not need the precise rate function I think the concentration result was already known Nov 8, 2021 at 22:17

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.