I will introduce a random matrix model for the statistical properties of the partial purity of a pure state in a bipartite quantum system at a finite fictitious temperature. This enables one to write the generating function for the cumulants, for both balanced and unbalanced bipartitions. It also unveils an unexpected feature of the system, namely the existence of two phase transitions, characterized by different spectra of the reduced density matrices. One of the critical phases is described by the statistical mechanics of 2-D quantum gravity, the other is a second-order phase transition.