[westoncb]

Hey, thanks a lot! Didn't expect such a great reply. I was still a bit fuzzy on a couple points, if you don't mind. I'm visiting another country and haven't found reliable wifi yet, hence slow reply time.

I think I get the scale invariant theory concept from a mathematical perspective, but I don't see why this would be the case: "Because you've got this considerable mixing of the two states, often "zooming in" is the same as, say, adjusting the proportion of liquid to gas"

Regarding random matrices, my understanding now is that given a random matrix of sufficient size, under a suitable definition of "random," if we look at the eigenvalue density function it's always going to be (roughly?) the same--we at least know it will be semi-circular. Further, there exist physical systems that can be modeled by random matrices; and there's a mapping between eigenvalues of the matrix and certain physical characteristics of the system. So, knowing the density function is always the same for these random matrices, we can assume certain shared characteristics of any systems that can be modeled by a random matrix.

In random matrices and phase transitions we would like to know how certain macroscopic parameters will behave, given some data like a matrix, or state of a phase transition. But, in both cases our starting data contains a lot of essentially irrelevant data that these theories prescribe a method for filtering out, since it assures us that knowledge of the symmetries involved are all that will matter.

Am I close? :)