What if we assume a finite universe instead? Contrary to what the post we’re discussing might suggest, this actually makes recurrence more reasonable. To show that every state of a finite universe recurs infinitely often, we only need to know one thing: that every state of the universe can be eventually reached from every other state.
Is this plausible? I’m not sure. The first objection that comes to mind is entropy: if entropy always increases, then we can never get back to where we started. But I seem to recall a claim that entropy is a statistical law: it’s not that it cannot decrease, but that it is extremely unlikely to do so. Extremely low probabilities do not frighten us here: if the universe is finite, then all such probabilities can be lower-bounded by some extremely tiny constant, which will eventually be defeated by infinite time.
But if the universe is infinite, this does not work: not even if the universe is merely potentially infinite, by which I mean that it can grow to an arbitrarily large finite size. This is already enough for the Markov chain in question to have infinitely many states, and my intuition tells me that in such a case it is almost certainly transient.
You are absolutely correct. If the number of states of the universe is finite, then as long as any state is reachable from any other state, then every state will be reached arbitrarily often if you wait long enough.
What if we assume a finite universe instead? Contrary to what the post we’re discussing might suggest, this actually makes recurrence more reasonable. To show that every state of a finite universe recurs infinitely often, we only need to know one thing: that every state of the universe can be eventually reached from every other state.
Is this plausible? I’m not sure. The first objection that comes to mind is entropy: if entropy always increases, then we can never get back to where we started. But I seem to recall a claim that entropy is a statistical law: it’s not that it cannot decrease, but that it is extremely unlikely to do so. Extremely low probabilities do not frighten us here: if the universe is finite, then all such probabilities can be lower-bounded by some extremely tiny constant, which will eventually be defeated by infinite time.
But if the universe is infinite, this does not work: not even if the universe is merely potentially infinite, by which I mean that it can grow to an arbitrarily large finite size. This is already enough for the Markov chain in question to have infinitely many states, and my intuition tells me that in such a case it is almost certainly transient.
You are absolutely correct. If the number of states of the universe is finite, then as long as any state is reachable from any other state, then every state will be reached arbitrarily often if you wait long enough.