Phase transitions of the p-spin model on pure Husimi lattices.

We consider the p-spin model with spin 1/2 on all pure Husimi lattices. Using an effective representation of the recursion relations, the phase transitions of the model on all pure Husimi lattices are investigated. First, the nonexistence of the second order phase transitions in the model on all pure Husimi lattices is proven exactly. Then the existence and properties of the first order phase transitions in a zero external magnetic field are studied in detail. An implicit polynomial equation for determining temperatures below which the paramagnetic and ferromagnetic phases coexist in the model on all pure Husimi lattices is found. In addition, an implicit equation for exactly determining the transition temperatures of the first order phase transitions in a zero external magnetic field on all pure Husimi lattices is derived and discussed.