Every mathematician is familiar with the result (due to Pólya) that for a random walk in a $d$-dimensional lattice, the probability $p(d)$ for returning to the origin eventually is $1$ if $d=1,2$, but $<1$ if $d>2$. There are also many discussions of this result on this site (such as Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1 discussing a proof).
What I have never seen is an intuitive explanation for the extremely counterintuitive result that for dimensions $1$ and $2$ the random walk returns to the origin almost surely but for higher dimensions it does not.
Does such an intuition exist?
I read the following intuitive statement somewhere: After N steps in any dimension, you are at average distance $sqrt(N) $ from the origin. In 3 dimensions therefore you can reach $N^{3/2}$ points, which is growing faster than N, so you cannot on average visit all the points. In 2d however this calculation gives that you can visit N points, which is (just) achievable.