One-sided large deviations for the ground-state energy of spin glasses

We describe the large deviations above its typical value of the maximal energy of a spin glass with +/-1 spins. Thanks to the relatively explicit description of the rate function we identify, we then show that the latter is asymptotically quadratic near its minimum if and only if an external magnetic field is present. The proof starts from a Parisi-type formula for the fractional moments of the partition function, which we then leverage to obtain the limit of the Laplace transform of the maximum energy. Using convex-duality arguments, we then rewrite this Laplace transform as a supremum over martingales, and thereby deduce the large-deviation principle with explicit rate function.