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 Phi(n,T)=TZ(n)/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 Phi(n,T) in the (n,T) plane computing in particular the stability of the replica-symmetric solution. At low temperatures we compute Phi(n,T) in series of n and tau=T(c)-T at high orders using the standard hierarchical ansatz and confirm earlier findings on the O(n(5)) scaling. We prove that the O(n(5)) 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.