When does a discrete-time random walk in $\mathbb{R}^n$ absorb the origin into its convex hull?

TL;DR

This paper investigates when a discrete-time random walk in high-dimensional space absorbs the origin into its convex hull, linking it to random matrix theory and providing probabilistic bounds on the walk's behavior.

Contribution

It introduces a novel connection between random walk absorption probabilities and random matrix intersection estimates, extending escape phenomena to broader classes of matrices.

Findings

High probability of $rac{ ext{pi}}{2}$-covering time being order $n$ for certain walks.

Extension of escape phenomena to broad classes of random matrices.

Exponential steps ($e^{Cn}$) suffice for the walk to absorb the origin in high dimensions.

Abstract

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon's escape theorem dealing with standard Gaussian matrices. The approach allows us to prove that with high probability, the $\pi/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we extend the "escape" phenomenon to a broad class of random matrices; as an application, we show that $e^{Cn}$ steps are sufficient for the standard walk on $\mathbb{Z}^n$ to absorb the origin into its convex hull with a high probability.