It is well known that a 2D random walk visits any square infinitely often almost surely, while a 3D random walk visits the origin finitely often almost surely. However, a 3D random walk does visit the line $L(\mathbb{Z}):=\{(0,0,k):k\in\mathbb{Z}\}$ infinitely often almost surely, as this practically reduces to a 2D random walk visiting the origin infinitely often almost surely. For any subset $S\subset\mathbb{N}$ of positive density, it also feels intuitively clear that the set $L(S):=\{(0,0,n):n\in S\}$ is visited infinitely often almost surely, as any time $L(\mathbb{Z})$ is visited, there is a positive probability that $L(S)$ is visited as well. Since the density of primes converges to $0$ very slowly, an interesting follow up question is thus the following.
Question: Is the set $L(\mathbb{P}):=\{(0,0,p):p\in\mathbb{N}\ \mbox{prime}\}$ visited infinitely often almost surely?
My guess is yes, but barely, in a way. Because I expect the set $\{(0,0,p,q):p,q\in\mathbb{N}\ \mbox{prime}\}$ is visited finitely often almost surely in a 4D random walk. Estimating the probability a 1D random walk is at the origin after $N$ steps as $N^{-1/2}$ and the probability of being prime after $N$ steps as $1/\log N$ gives either the divergent sum $\sum 1/N\log N$ or the convergent sum $\sum 1/N\log^2N$. I am wondering if this argument can be made precise and whether there is more literature on those sets which are visited infinitely often almost surely.