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 Gärtner-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 calculation that the rate function is infinite for fluctuations of the free energy above its typical value, corresponding to an anomalous, superextensive suppression of rare fluctuations. We extend this calculation to nonzero 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.