Research on replica symmetry breaking in sparsely connected spin glass models

本研究的目的是探讨在稀疏连接的伊辛自旋玻璃模型中观察到的复制对称性破缺（RSB）现象。特别是，我们寻求一个一般的表达的de Almeida无（AT）的条件，信号的局部不稳定性RSB和检查的稀疏连接模型的RSB相位的属性。主要研究结果如下：1.利用置信传播算法计算自旋玻璃极化率，得到AT条件的一般表达式。我们还发现，AT条件是由一个随机过程，这是一个大的偏差统计，并开发了一种方法来评估大偏差率函数的随机过程的特点.提出了一种有效搜索Bethe逼近解的算法。我们使用该算法对Bethe自由能进行了Hessian分析，发现当外场相对较小时，尽管临界点的数值确定是困难的，但Bethe近似的解很可能保持预期的有限尺寸标度关系.对于马尔可夫链蒙特卡罗为基础的评估的临界点，我们研究了三个措施，主要特征值的磁化率矩阵A，自旋玻璃磁化率矩阵ASG和自旋玻璃磁化率XSG的属性。我们发现对于A和ASG，由于有限尺寸效应，一定的偏差修正是必不可少的，尽管ASG在数据准确性方面是最好的.用复型法研究了消失温度范围内的性能。分析表明，基态具有一步RSB解的特征。这一性质与全连通模型相反，全连通模型的零温度状态通常由全RSB解描述。

The objective of this research is to explore the replica symmetry breaking (RSB) phenomena observed in sparsely connected spin glass models of Ising spins. In particular, we seek for a general expression of the de Almeida-Thouless (AT) condition which signals the local instability for RSB and examine properties of the RSB phase of the sparsely-connected models. The obtained results are summarized as follows.1. A general expression of the AT condition was obtained by evaluation of the spin glass susceptibility utilizing the belief propagation algorithm. We also found that the AT condition is determined by a stochastic process that is subject to a large deviation statistics, and developed a methodology to assess the large deviation rate function that characterizes the stochastic process.2. An algorithm to efficiently search Bethe approximation solutions is developed. We performed the Hessian analysis of the Bethe free energy using this algorithm and found that an expected finite size scaling relation is likely to hold for solutions of the Bethe approximation when the external field is relatively small although numerical determination of the critical point is difficult.3. For Markov Chain Monte Carlo-based assessment of the critical point, we examined properties of three measures, the principal eigen value of the susceptibility matrix A, that of the spin glass susceptibility matrix ASG and the spin glass susceptibility XSG. We found that for A and ASG, certain bias corrections are indispensable due to finite-size effect although ASG is most preferred in terms of accuracy of data.4. Properties in the limit of vanishing temperature was examined by using the replica method. The analysis indicates that the ground states are characterized by the one step RSB solution. This property is in contrast with that of the fully connected models, the zero temperature states of which are generally described by the full RSB solution.