A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. converges point-wise). This is because in order for a phase change to occur, the partition function must converge to two different continuous functions in different regions of the phase plane; and this can only happen if the number of terms in the partition function is infinite.
It might be possible, using the analogy with phase changes, to make some general statements about partition functions which converge non-uniformly to two different continuous functions. This is interesting to me because point-wise convergence is a very weak condition, and the analogy with phase changes seems to suggest a path towards a deeper understanding of this form of convergence.
One of the most interesting subjects in the statistical mechanics of phase transitions is the so-called "renormalization group" (which is not a group) and the critical exponents it predicts (these are non integer numbers which are associated with a "critical point" in the phase plane). Another powerful result from renormalization group theory is the concept of universality, in which the thermodynamic variables of the system are symmetric across all length scales at a certain critical point. Universality is directly related to the critical exponents and critical point behavior, but as is fairly standard for theoretical physics, the mathematics used is non-rigorous (I recommend the book "scaling and renormalization in statistical physics" by Cardy)
For the point-wise convergence of a partition function, can we put the foundations of critical exponents on a rigorous mathematical basis; in the sense that we can somehow predict their values from properties (such as the partial derivatives) of the partition function?
