Intuition for when the replica trick should work and why it works

Question

I am a graduate student in mathematics working in probability (without a very good background in physics honestly) and I've started to see arguments based on computations derived from the replica trick. I understand that it is non-rigorous but it appears that a decent number of the solutions one obtains using this trick can be made rigorous by other (often fairly involved) methods. I would like to work on developing a better heuristic approach to problems, so I would like to understand when I can reasonably expect this type of argument to give reasonably accurate predictions. Perhaps more importantly for me, is there a nice characterization of physical situations when it is clear that this trick should fail?

Links to introductory papers, worked elementary examples, or hand waiving arguments about why this should work and when it shouldn't would all be appreciated. I've glanced at the first few PDFs that appear when you google "Replica Trick" so no need to link me to lmgtfy :)

This post imported from StackExchange Physics at 2017-02-05 11:31 (UTC), posted by SE-user Chris Janjigian