Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
Phys Rev Lett. 2015 Jan 16;114(2):028103. doi: 10.1103/PhysRevLett.114.028103.
We consider statistical-mechanics models for spin systems built on hierarchical structures, which provide a simple example of non-mean-field framework. We show that the coupling decay with spin distance can give rise to peculiar features and phase diagrams much richer than their mean-field counterpart. In particular, we consider the Dyson model, mimicking ferromagnetism in lattices, and we prove the existence of a number of metastabilities, beyond the ordered state, which become stable in the thermodynamic limit. Such a feature is retained when the hierarchical structure is coupled with the Hebb rule for learning, hence mimicking the modular architecture of neurons, and gives rise to an associative network able to perform single pattern retrieval as well as multiple-pattern retrieval, depending crucially on the external stimuli and on the rate of interaction decay with distance; however, those emergent multitasking features reduce the network capacity with respect to the mean-field counterpart. The analysis is accomplished through statistical mechanics, Markov chain theory, signal-to-noise ratio technique, and numerical simulations in full consistency. Our results shed light on the biological complexity shown by real networks, and suggest future directions for understanding more realistic models.
我们考虑基于层次结构的自旋系统的统计力学模型，这为非平均场框架提供了一个简单的例子。我们表明，自旋距离的耦合衰减可以产生比平均场对应物更丰富的奇特特征和相图。具体来说，我们考虑了 Dyson 模型，它模拟了晶格中的铁磁性，并且我们证明了在有序状态之外存在许多亚稳态，这些亚稳态在热力学极限下变得稳定。当层次结构与学习的赫布规则耦合时，这种特征得以保留，从而模拟了神经元的模块化结构，并产生了一个联想网络，能够执行单一模式检索以及多模式检索，这主要取决于外部刺激和距离上的相互作用衰减速率；然而，这些新兴的多任务特征相对于平均场对应物降低了网络容量。该分析通过统计力学、马尔可夫链理论、信噪比技术和完全一致的数值模拟来完成。我们的研究结果揭示了真实网络所表现出的生物复杂性，并为理解更现实的模型提供了未来的方向。
