Free energy in multi-species mixed $p$-spin spherical models

Understand

We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions.

• The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function.
• Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption.
• In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space.