Convergence of the Free Energy for Spherical Spin Glasses

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. We first relate the free energy on the sphere of dimension N+ M to a free energy defined on the product of spheres in dimensions N and M to which we then apply the interpolation method. This yields an approximate super-additivity which is sufficient to prove the convergence.