You can guarantee that the probability is the theoretical minimum with a bijection. I think that would be 2^-N since it's just the case where everything's on a maximum length cycle, but I haven't thought about it hard enough to be completely certain.

A good hash function intentionally won't hit that level, but it should be close enough not to matter with 64 bit pointers. 32 bits is small enough that I'd have concerns at scale.