TY - JOUR

T1 - Convergence of the Free Energy for Spherical Spin Glasses

AU - Subag, Eliran

N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/11

Y1 - 2022/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85138300869&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s10955-022-02988-2

DO - https://doi.org/10.1007/s10955-022-02988-2

M3 - مقالة

SN - 0022-4715

VL - 189

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 2

M1 - 29

ER -