Convergence of the free energy for spherical spin glasses

Understand

We prove that the free energy of any spherical mixed $p$-spin model converges as the dimension $N$ tends to infinity.

• While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the well-known Guerra-Toninelli interpolation method.
• The latter was invented for models with Ising spins to prove that the free energy is super-additive and therefore (normalized by $N$) converges.
• In the spherical case, however, the configuration space is not a product space and the interpolation cannot be applied directly.