Phase diagram and large deviations in the free energy of mean-field spin glasses

We consider the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean-field model, i.e., the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing Φ (n,T) =T Zn ̄ /n, i.e., the average value of the partition function to the power n as a function of n. We study in full details the phase diagram of Φ (n,T) in the (n,T) plane computing in particular the stability of the replica-symmetric solution. At low temperatures we compute Φ (n,T) in series of n and τ= Tc -T at high orders using the standard hierarchical ansatz and confirm earlier findings on the O (n5) scaling. We prove that the O (n5) scaling is valid at all orders and obtain an exact expression for the coefficient in term of the function q (x). Resumming the series we obtain the large deviations probability at all temperatures. At zero temperature the analytical prediction displays a remarkable quantitative agreement with the numerical data. A similar computation for the simpler spherical model is also performed and the connection between large and small deviations is discussed.