Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the singular values. What's the joint distribution of $$\sigma_1 - \sigma_2, \sigma_2 - \sigma_3, \sigma_3 - \sigma_4, \ldots?$$