Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \Sigma_m V_m^T$ as the singular value decomposition, where $V_m \in \mathbb{R}^{n \times n}$ are the right singular vectors.

What can be said about $V_m$ as $m \to \infty$? Is anyone aware of literature or techniques discussing this?