The laws of physics don't change, or if current ideas hold water, don't change a whole lot. Some things aren't possible. If it's possible, then given an infinite amount of time and circumstances it's bound to happen sometime, somewhere. But such conditions don't exist anyway.
I don't know if "there is no such thing as random in infinity" has a lot of meaning. Infinity isn't something you can be in. It's like saying "there is no such thing as orange in psychiatry". Psychiatry has nothing to do with colours, and infinity has nothing to do with patterns.