>  Schwarzchild radius is not dependent on density, only on the amount of
> mass.

That's only true for black holes. For normal matter it varies by density,
or is posed as the answer to the question, "If I have some mass and turn
it into a black-hole, how big will it be". That's not to say that the 
universe
is all in a black hole, but if the mass of the universe is infinite it falls 
inside
it's Schwarzschild radius, since rs = 2Gm/c^2 ... and infinity divided by
a large number is still infinity.

>> If the universe is infinite, with infinite mass, then at some distance
>> the same patterns of atoms would be there. There are only a fixed
>> number of atom types, all the same laws of physics apply, and even
>> if the number of atom arrangements on a planet is very huge, it's not
>> infinite, so it must be repeated at some point in an infinite universe.
>
>  I call this the shakespeare-monkey fallacy.
>
>  Just because there's an infinite amount of data doesn't mean that every
> possible finite pattern appears, nor that a given finite pattern appears
> an infinite number of times.
>
>  Even if the universe had an infinite amount of mass (the current
> consensus is that this is not so), only a finite amount of it may be
> forming atoms. The rest can be eg. singularities, dark energy, dark
> matter, etc. (And even if there's an infinite amount of atoms, it doesn't
> necessarily mean that that certain patterns must repeat.)

It's not a fallacy, it's a paradox, there's a difference. It's not possible
to predict random events, but it's still possible to know that they will
occur. For example if you use a simple coin toss, and toss the coin
forever, the distribution will be 50/50, so you can say with 100%
confidence that sometime in the future there will be another "heads",
yet at the same time you don't know at all if the next toss will be
a "heads".

If the universe is finite then such super low probability odds as
a parallel earth would never occur, but if the universe is infinite,
then not only will there be parallels, but there will be an infinite number
of them.  I'm not convinced that the universe is infinite, but
astronomers haven't seen any uniform expanding or contracting that
is predicted by general relativity, they do see bubble like areas where
it looks like some repulsive force is at play. And there's the dark matter.
General relativity didn't predict most of these observations, in some
ways it's great, like Newton's theory, in others ways it falls short.

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