Large deviations of the free energy in the p-spin glass spherical model

We investigate the behavior of the rare fluctuations of the free energy in the p-spin spherical model, evaluating the corresponding rate function via the G\"artner-Ellis theorem. This approach requires the knowledge of the analytic continuation of the disorder-averaged replicated partition function to arbitrary real number of replicas. In zero external magnetic field, we show via a one-step replica symmetry breaking (1RSB) calculation that the rate function is infinite for fluctuations of the free energy above its typical value, corresponding to an anomalous, super-extensive suppression of rare fluctuations. We extend this calculation to non-zero magnetic field, showing that in this case this very large deviation disappears and we try to motivate this finding in light of a geometrical interpretation of the scaled cumulant generating function.