论文标题
球形自旋玻璃的自由能的收敛性
Convergence of the free energy for spherical spin glasses
论文作者
论文摘要
我们证明，随着尺寸$ n $倾向于无限，任何球形混合$ p $ spin模型的自由能都会收敛。虽然收敛是巴黎公式的结果，但我们给出的证明与公式无关，并使用了众所周知的Guerra-Toninelli插值方法。后者是针对带有Ising旋转的模型发明的，以证明自由能是超级加化的，因此（由$ n $进行标准化）收敛。但是，在球形情况下，配置空间不是产品空间，无法直接应用插值。我们首先将尺寸$ n+m $领域的自由能与尺寸$ n $和$ m $的产品定义的自由能联系起来，然后我们将其应用插值方法。这产生了近似的超级促进性，足以证明收敛性。
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.
