What does become weird?

Consider normal distribution. n-dimensional vectors with independent N(0,1) with expectation 0.

When n increases the mass of the probabilities are around a sphere of radius n^0.5, nowhere near the origin.

True, a huge volume with low density will contain more mass than a small volume with large density. But we can say that it gets “concentrated” far from the origin only in the same sense that Japan’s population is “concentrated” outside of Tokio…