We focus on spherical spin glasses whose Parisi distribution has support of the form $[0,q]$. For such models we construct paths from the origin to the sphere which consistently remain close to the ground-state energy on the sphere of corresponding radius. The construction uses a greedy strategy, which always follows a direction corresponding to the most negative eigenvalues of the Hessian of the Hamiltonian. For finite mixtures $\nu(x)$ it provides an algorithm of time complexity $O(N^{{\rm deg}(\nu)})$ to find w.h.p. points with the ground-state energy, up to a small error. For the pure spherical models, the same algorithm reaches the energy $-E_{\infty}$, the conjectural terminal energy for gradient descent. Using the TAP formula for the free energy, for full-RSB models with support $[0,q]$, we are able to prove the correct lower bound on the free energy (namely, prove the lower bound from Parisi's formula), assuming the correctness of the Parisi formula only in the replica symmetric case.

中文翻译：

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遵循全 RSB 球面自旋玻璃的基态

我们专注于其帕里西分布支持 $[0,q]$ 形式的球形自旋眼镜。对于这样的模型，我们构建了从原点到球体的路径，这些路径始终保持接近相应半径球体上的基态能量。该构造使用贪婪策略，该策略始终遵循与哈密顿量的 Hessian 的最负特征值相对应的方向。对于有限混合 $\nu(x)$，它提供了一种时间复杂度 $O(N^{{\rm deg}(\nu)})$ 的算法，以找到具有基态能量的 whp 点，直到一个小的错误。对于纯球形模型，相同的算法达到能量 $-E_{\infty}$，即梯度下降的推测终端能量。使用自由能的 TAP 公式，对于支持 $[0,q]$ 的全 RSB 模型，