Limit formulas for metric measure invariants and phase transition property

We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition property for a sequence of metric measure spaces or pyramids, and find some examples of symmetric spaces of noncompact type with the phase transition property. We also give a simple proof of a theorem in Funano and Shioya (Geom Funct Anal 23(3):888–936, 2013) on the limit of an N-Lévy family.