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

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 Sn−1Sn−1. In this way, the case of a discretized Brownian motion is related to Gordon’s escape theorem dealing with standard Gaussian matrices. We show that for the random walk BMn(i),i∈NBMn(i),i∈N, the convex hull of the first CnCn steps (for a sufficiently large universal constant CC) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the π/2π/2-covering time of certain random walks on Sn−1Sn−1 is of order nn. For certain spherical simplices on Sn−1Sn−1, we prove an extension of Gordon’s theorem dealing with a broad class of random matrices; as an application, we show that CnCn steps are sufficient for the standard walk on ZnZn to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant c>1c>1, the convex hull of the nn-dimensional Brownian motion conv{BMn(t):t∈[1,cn]}conv⁡{BMn(t):t∈[1,cn]} does not contain the origin with probability close to one.