Convergence of the Free Energy for Spherical Spin Glasses

• Published in: Journal of statistical physics Vol. 189; no. 2
• Main Author: Subag, Eliran
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2022; Springer; Springer Nature B.V
• Subjects: Convergence; Free energy; Interpolation; Ising model; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Spin glasses; Statistical Physics and Dynamical Systems; Theoretical; Spherical spin glass; Super-additivity; Free energy; Guerra–Toninelli interpolation
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-022-02988-2

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.