Time required for the convex hull of a random walk in $\mathbb{R}^n$ to absorb the origin: A random matrix approach

Addressing a question of Benjamini, considered previously by Eldan, we estimate the number of steps required by a random walk in $\mathbb{R}^n$ to include the origin in its convex hull. Further, we show that with high probability the $\pi/2$-covering time for certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. To obtain these results, we prove some general statements about random matrices, closely related to Gordon's escape theorem.