This question is partially inspired by following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional statistics.

Instead of asking about surprising results in high dimensions, I will ask what kind of results that holds in low dimensions fails to hold in a higher dimension. And what is its relation with other mathematical branch?

(By high dimensional statistics we usually refer to a high dimension covariate space instead of response space. For example, in a regression setting $Y=f(X)$ we tend to say a problem is of high dimension if $dim\mathcal{X}\gg dim\mathcal{Y}$)

For one simplest example, we know that James-Stein estimator performs better than maximum likelihood estimator in terms of $L^2$ norm when the dimension $dim\mathcal{X}=d\geq 3$; and it turns out to be an equivalent statement that a symmetric random walk in $\mathcal{X}=\mathbb{R}^d$ is transient when $d\geq 3$ via an infinitely divisible stochastic process.

Are there other such examples that can relate high dimensional phenomena in statistics?