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We generalize the observable diameter and the separation distance for metricmeasure spaces to those for pyramids, and prove some limit formulas for theseinvariants for a convergent sequence of pyramids. We obtain variousapplications of our limit formulas as follows. We have a criterion of the phasetransition property for a sequence of metric measure spaces or pyramids, andfind some examples of symmetric spaces of noncompact type with the phasetransition property. We also give a simple proof of a theorem by Funano-Shioyaon the limit of an $N$-L\'evy family.

Limit formulas for metric measure invariants and phase transition property