TITLE: "Phase change" of a purely mathematical system
QUESTION [9 upvotes]: Every so often I hear people talking about "phase transitions" in purely mathematical or computer-science contexts, where there is no physics in sight. Today, for example, I heard some people talking about "phase transitions" when coloring edges in random graphs.
Is this just a simple reuse of the physics term, or is there a more general mathematical concept of a "phase change", of which physical phase changes are one instance? If so, what are some illustrative purely mathematical/nonphysical examples?

REPLY [2 votes]: I don't have the reputation to post a comment but ...
"Where the Really Hard Problems Are" makes a critical error relative to where the really hard problems are.
Problems in the narrow "boundary region" can be easily moved to the "over constrained region" by adding n new variables and 6n new clauses. These clauses need only involve those n varaibles, but need not. This is a polynomial transformation that has no effect on the solution to the original problem.
I would suggest that using "phase change" in this context might be improper.