It was proved by Michel Talagrand in  that the Parisi formula for the free energy in the Sherrington-Kirkpatrick model is differentiable with respect to inverse temperature parameter. We present a simpler proof of this result by using approximate solutions in the Parisi formula and give one example of application of the differentiability to prove non self-averaging of the overlap outside of the replica symmetric region.
"On differentiability of the Parisi formula." Electron. Commun. Probab. 13 241 - 247, 2008. https://doi.org/10.1214/ECP.v13-1365