The behavior of a random walk in a high dimensional space can be counter-intuitive. If you take the random walk trajectory and then perform principal components analysis on it, it turns out more than half of the variance is along a single direction. More than 80% is along the first two principal components.

To make matters even more surprising, if you project the random walk trajectory down into these PCA subspaces they are no longer random at all. Instead the trajectory traces a Lissajous curve. (For example see figure 1 of this paper: https://proceedings.neurips.cc/paper/2018/file/7a576629fef88...)