Construction of pure states in mean field models for spin glasses
Construction of pure states in mean field models for spin glasses
复制标题
自旋玻璃平均场模型中纯态的构建
DOI：
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发表时间：
2010
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通讯作者：
M. Talagrand
中科院分区：
文献类型：
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作者：
M. Talagrand
如果自旋玻璃的平均场模型在满足扩展的Ghirlanda-Guerra恒等式的意义上是一般的，并且如果重叠定律在其支持的最大点q*处有一个点质量，我们证明可以将构型空间分解为一个集合序列（Ak），使得两个构型的重叠一般地等于q*当且仅当它们属于同一个集合Ak。对于重叠的研究，每个集合Ak都可以用一个点来代替。结合Panchenko最近的结果（Ghirlanda-Guerra身份和超韵律之间的联系）。这证明了如果重叠只取有限多个值，就会发生超对称。我们基于简单不等式和一个平均论证给出了这个结果的初等自包含证明。
If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda–Guerra identities, and if the law of the overlaps has a point mass at the largest point q* of its support, we prove that one can decompose the configuration space into a sequence of sets (Ak) such that, generically, the overlap of two configurations is equal to q* if and only if they belong to the same set Ak. For the study of the overlaps each set Ak can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda–Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.