User blog comment:Deedlit11/The slow-growing hierarchy and other hierarchies/@comment-25418284-20130317191338/@comment-157.193.53.9-20130620120634

Yes, that is the paper I referred to. There is also a paper by Friedman and Sheard in APAL which comes to similar conclusions regarding hierarchies. Regarding phase transitions a possible start is:

MR2500089 (2010f:03049) Omri, Eran; Weiermann, Andreas Classifying the phase transition threshold for Ackermannian functions. Ann. Pure Appl. Logic 158 (2009), no. 3, 156–162.

But this was just the start of the research project. The other results are still unpublished.