Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobi technique
Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobi technique
复制标题
通过汉密尔顿-雅可比技术在平均场自旋玻璃中实现复制对称性破缺
DOI：
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复制
发表时间：
2010
期刊：
影响因子：
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通讯作者：
F. Guerra
中科院分区：
文献类型：
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作者：
A. Barra;A. D. Biasio;F. Guerra
在过去的几年里，通过来自物理直觉和计算机模拟的洞察力的共同努力，以及严格的数学方法的开发，平均场Sherrington-Kirkpatrick自旋玻璃模型的主要特征已经牢固地建立起来。特别地，用涉及泛函序参数的变分原理证明了自由能无限体积极限及其Parisi表达式的存在唯一性。对于无限体积的状态，即使是预期的超度规性质，似乎也接近于完全的证明。这个模型和相关模型的主要结构特征是由Parisi多年前发现的自发性复制对称破缺(RSB)现象。通过扩展我们以前的工作，本文的目的是研究一个一般的框架，其中副本对称性破缺嵌入到一种Hamilton-Jacobi型的力学方案中。这里，‘时间’变量的模拟是表征相互作用强度的参数，而‘空间’变量定量地排除了破坏的复制对称模式。从假设退火态或复制对称性的简单情况出发，我们建立了一个随着空间变量的增加而增加的动力学系统的级数，这使得我们可以随着对称性破缺程度的增加而减弱Hamilton-Jacobi方程中势的影响。这种新的机制允许我们机械地计算出一般的K步RSB解，相对于复制技巧有不同的解释，并且很容易揭示它们的存在或唯一性等性质。
During the last few years, through the combined effort of the insight coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean-field Sherrington–Kirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite-volume limit for the free energy, and its Parisi expression, in terms of a variational principle involving a functional order parameter. Even the expected property of ultrametricity, for the infinite-volume states, seems to be near to a complete proof. The main structural feature of this model, and related models, is the deep phenomenon of spontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. By expanding on our previous work, the aim of this paper is to investigate a general framework, where replica symmetry breaking is embedded in a kind of mechanical scheme of the Hamilton–Jacobi type. Here, the analog of the ‘time’ variable is a parameter characterizing the strength of the interaction, while the ‘space’ variables rule out quantitatively the broken replica symmetry pattern. Starting from the simple cases, where annealing is assumed, or replica symmetry, we build up a progression of dynamical systems, with an increasing number of space variables, which allow us to weaken the effect of the potential in the Hamilton–Jacobi equation as the level of symmetry breaking is increased. This new machinery allows us to work out mechanically the general K-step RSB solutions, in a different interpretation with respect to the replica trick, and easily reveals their properties such as existence or uniqueness.