Different arrangements would do it, or you could think of it very loosely as the "if you made different decisions in each" kind of thing. Events/history is different, essentially.

On your idea of a "random function" yes, pretty much everything at a fundamental level appears to be probabilistic to some level. Quantum theory cannot in most cases predict the exact outcome of an experiment, just the probability of different outcomes.

You can combine the same atoms into different larger elements. It's like that. The same physical laws, but the combinations are infinite (or at least uncountable).

The other concept is that the laws are different, but not true infinite combinations within reach.

Or both.