The convex structure of the Parisi formula for multi-species spin glasses

We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on ${\mathbb{R}}_{+}^D$. We show here that one can transform this representation into a supremum over all probability measures on ${\mathbb{R}}_{+}^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using convex-duality arguments, we also obtain a new representation of the free energy as an infimum over martingales in a Wiener space.