Polya’s recurrence theorem sheds some light here. Running forever in the infinite meadow, the blindfolded child is guaranteed to hit the post. But if the child’s moving through a higher-dimensional space, his chance of hitting the post is very small unless the post starts out nearby. A fish swimming randomly, forever, through an ocean of infinite breadth and depth, has only a 36% chance of ever returning to its starting position. Higher dimensionality helps us avoid random hazards, but also prevents us from finding random benefits.
It seems worth noting that, though humans do live in a very high-dimensional configuration space, the “hazards” we worry about also live in that same space, and as such may not be such easily avoidable objects as the points discussed by Polya’s recurrence theorem. (An infinite line in three-dimensional space, for example, is analogous to a point on a two-dimensional plane, and is likewise guaranteed to be hit by a sufficiently long random walk.)
It seems worth noting that, though humans do live in a very high-dimensional configuration space, the “hazards” we worry about also live in that same space, and as such may not be such easily avoidable objects as the points discussed by Polya’s recurrence theorem. (An infinite line in three-dimensional space, for example, is analogous to a point on a two-dimensional plane, and is likewise guaranteed to be hit by a sufficiently long random walk.)