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?