短程、稀释及Lévy自旋玻璃中的无序混沌
Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses
AI总结：
本文利用Hermite谱方法将位点无序混沌从EA短程自旋玻璃模型推广到混合偶p-自旋短程模型、稀释混合p-自旋模型及稳定指数在(1,2)的Lévy模型，并通过Fourier-Hermite系数代数方程确定重叠函数中有贡献的系数。
摘要：
在近期的一项突破性工作[arXiv:2301.04112]中，Chatterjee利用Hermite谱方法证明了Edwards-Anderson（EA）短程自旋玻璃模型中的位点无序混沌。本文通过将位点无序混沌的有效性推广到三个相关的自旋玻璃模型，展示了这种Hermite谱方法的进一步用途。第一个模型称为混合偶p-自旋短程模型，是EA模型的推广，其底层图是一个确定性的有界度超图，由具有偶数个顶点的超边组成。第二个模型是稀释混合p-自旋模型，允许超边同时具有奇数和偶数个顶点。对于这两个模型，我们的结果在一般对称无序分布下成立。我们论证的主要新颖之处在于，针对双自旋关联函数的Fourier-Hermite级数系数，使用了一个初等代数方程。它使我们能够推导出必要的几何条件，以确定重叠函数中有贡献的系数，这在精神上与[arXiv:2301.04112]中关键的引理1相同。最后，我们还在稳定指数为$α\in (1, 2)$的Lévy模型中建立了无序混沌。
英文摘要：
In a recent breakthrough [arXiv:2301.04112], Chatterjee proved site disorder chaos in the Edwards-Anderson (EA) short-range spin glass model utilizing the Hermite spectral method. In this paper, we demonstrate the further usefulness of this Hermite spectral approach by extending the validity of site disorder chaos in three related spin glass models. The first, called the mixed even $p$-spin short-range model, is a generalization of the EA model where the underlying graph is a deterministic bounded degree hypergraph consisting of hyperedges with even number of vertices. The second model is the diluted mixed $p$-spin model, which is allowed to have hyperedges with both odd and even number of vertices. For both models, our results hold under general symmetric disorder distributions. The main novelty of our argument is played by an elementary algebraic equation for the Fourier-Hermite series coefficients for the two-spin correlation functions. It allows us to deduce necessary geometric conditions to determine the contributing coefficients in the overlap function, which in spirit is the same as the crucial Lemma 1 in [arXiv:2301.04112]. Finally, we also establish disorder chaos in the Lévy model with stable index $α\in (1, 2)$.
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