The free energy in a multi-species Sherrington.Kirkpatrick model

The authors of [Ann. Henri Poincaré 16 (2015) 691.708] introduced a multi-species version of the Sherrington.Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra.s replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda.Guerra identities and showing that they force the overlaps within species to be completely determined by the overlaps of the whole system.