We consider statistical-mechanical 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 that their mean-field counterpart. In particular, we consider the Dyson model, mimicking ferromagnetism in lattices, and we prove the existence of a number of meta-stabilities, beyond the ordered state, which get 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 both as a serial processor as well as a parallel processor, 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, graph theory, signal-to-noise 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.
A Bifurcation Theory for the Equilibria of Modern Hopfield Networks
Max-Planck-Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Gurdon Institute, University of Cambridge（剑桥大学格登研究所）
Existence of a Phase Transition in the One-Dimensional Ising Spin Glass Model with Long-Range Interactions on the Nishimori Line
Graduate School of Information Sciences, Tohoku University（东北大学情报科学研究院）；School of Computing, Institute of Science Tokyo（东京科学大学理工学院）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Research and Education Institute for Semiconductors and Informatics, Kumamoto University（熊本大学半导体与信息学研究与教育学院）；Sigma-i Co., Ltd.（Sigma-i有限公司）
