混合偶$p$ -自旋模型的能量图景
On the energy landscape of the mixed even $p$-spin model
摘要：
研究了具有Ising自旋组态的混合偶$p$ -自旋模型的能量图景.我们表明，对于任何给定的能量水平之间的零和最大的能量，以压倒性的概率存在指数许多不同的自旋配置，使他们的能量保持在这个能级附近。此外，它们的磁化强度和重叠度都集中在一些固定的常数附近。特别地，在最大能量的水平上，我们证明了哈密顿量呈现指数的许多正交峰。这改进了Chatterjee和Ding-Eldan-Zhai的结果，其中前者建立了正交峰数目的对数大小，而后者证明了多项式大小。我们的第二个主要结果得到了零温和任意外场下的无序混沌。作为一个副产品，这意味着当外场消失时，最大能量的波动是超集中的，并且当外场存在时服从高斯极限定律。
英文摘要：
We investigate the energy landscape of the mixed even $p$-spin model with Ising spin configurations. We show that for any given energy level between zero and the maximal energy, with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level. Furthermore, their magnetizations and overlaps are concentrated around some fixed constants. In particular, at the level of maximal energy, we prove that the Hamiltonian exhibits exponentially many orthogonal peaks. This improves the results of Chatterjee and Ding-Eldan-Zhai, where the former established a logarithmic size of the number of the orthogonal peaks, while the latter proved a polynomial size. Our second main result obtains disorder chaos at zero temperature and at any external field. As a byproduct, this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a Gaussian limit law when the external field is present.
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