We know that the symmetric random walk on $\mathbb{Z}^d, \, d \geq 3$ is transient. However, if we looked at each coordinate separately, we would see a lazy symmetric random walks on $\mathbb{Z}$, hence recurrent. Therefore, the whole process is transient, but its "sub-processes" are recurrent. Moreover, if we look at any pair of coordinates, these sub-process should be recurrent too.
Are these observations correct? If so, do you have an explanation or proof based on an intuitive idea?
Here is one attempt for an intuitive explanation for $d=3$.
You have Alice, Bob and Charles walking symmetric random walks on integers. They walk along the directions $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$. You roll a fair three sided die to decide who is walking next. Hence, the only place they can "meet" is on the origin.
As the process goes on, Alice strays away from the origin, but always come back. However, when she does, either Bob or Charles is already far from the origin. It is just too unlikely for both to be there. You proceed to ask her some questions:
Q: How often do you see Bob? A: It takes a long time, but I always end up meeting him.
Q: How often do you see Charles? A: It takes a long time, but I always end up meeting him.
Q: How often do you see Bob and Charles at the same time? A: I met both sometimes, but I think our group will never meet again.