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Phase diagram for the tap energy of the $p$-spin spherical mean field spin glass model

We solve the Thouless-Anderson-Palmer (TAP) variational principle associated to the spherical pure $p$-spin mean field spin glass Hamiltonian and present a detailed phase diagram. In the high temperature phase the maximum of variational principle is the annealed free energy of the model. In the low temperature phase the maximum, for which we give a formula, is strictly smaller. The high temperature phase consists of three subphases. (1) In the first phase $m=0$ is the unique relevant TAP maximizer. (2) In the second phase there are exponentially many TAP maximizers, but $m=0$ remains dominant. (3) In the third phase, after the so called dynamic phase transition, $m=0$ is no longer a relevant TAP maximizer, and exponentially many non-zero relevant TAP solutions add up to give the annealed free energy. Finally in the low temperature phase a subexponential number of TAP maximizers of near-maximal TAP energy dominate.