Results of Statistical Mechanics first obtained by formal mathematical methods I have a question that seems natural in Physics and Mathematics mainly in Statistical Mechanics of Equilibrium.
Results that are proven by formal mathematical methods that were already seem intuitive results and experimentally verified when proofs by mathematically rigorous methods were obtained.
In many fields of science, most notably physics, there are many historical examples that refute this view.
An example of this type that I know is the General Theory of Relativity and GPS (the original English acronym for Global Positioning System). It is possible that one may disagree. Without explaining in detail the formulas of contraction / dilation of spacetime which is obtained formally Theory of Relativity are used to "sync" properly watch each of the GPS satellites with the clocks of a point on Earth. For more see here and here.
However, I can not get an explicit  example in Statistical Mechanics. That is, would an example of resutado first obtained by formal mathematical methods and was discovered by experimental means later.
Question: There is a exemple of same result in  Statistical Mechanics that was first discovered by "theoretical mathematical methods" and only later confirmed experimentally ?
Question: Some example explicit or reference for the Ising model?
 A: I think that the most prominent example of "prediction before observation" in statistical physics is the Bose-Einstein condensate.
It was predicted in ~1925 by, well, Bose and Einstein, obviously. Then after more than ten years it was proposed as an explanation for superfluidity and superconductivity. And the actual BEC of atoms (as a new state of matter) was obtained only in 1995.
A: Off the top of my head, the example I can think of is the whole work that Boltzmann did. He based his entire theory of statistical mechanics on the concept of indivisible particles (i.e. that all matter is made up out of atoms). Doing this, his theory (using theoretical mathematical methods as you said) was able to predict how the atoms determine the visible properties matter. And remember, in the time of Boltzmann if you believed in atoms you were labelled a crackpot. Only later with Einstein's work on Brownian motion, Perrin's studies on colloidal suspensions and Rutherford's experiments people started to acknowledge the reality of the atom. I think this can be considered a prediction (and a fundamental one) before experimental discovery. 
A: A famous example from celestial mechanics: The original calculation of the orbit of Uranus http://en.wikipedia.org/wiki/Neptune indicated that the planet was being perturbed by some as yet unknown object in space. This formal series solution led directly to the discovery of the planet Neptune. 
