作为可精确求解模型的图on自旋系统
Graphon Spin Systems as Exactly Solvable Models
AI总结：
本研究将伊辛模型构建于图on之上，推导其平均场极限与相变精确结果，以三类随机网络验证，兼具可分析性与适配性，为复杂网络统计物理提供新视角。
AI中文摘要：
图on是用于描述收敛图族渐近行为的可测函数，最初由组合学与图论问题驱动，现已在网络上动力学过程的建模与分析中得到大量应用。本研究使用图on在收敛图序列上构建伊辛模型，这类序列包含诸多应用中常见的网络拓扑。我们推导了所得模型的平均场极限，并获得这类系统相变的精确结果；具体而言，证明了图on上伊辛模型的临界温度由与图极限相关的希尔伯特-施密特算子的本征值决定，对诸多重要网络拓扑，这些本征值可显式计算。我们以三类代表性随机网络模型说明所得结果：埃尔德里奇-雷尼（Erdős-Rényi）、小世界及幂律网络；在小世界情形中，我们证明了铁磁相与反铁磁相的相变，以及自由能局部极小值的共存，后者产生多稳态，这一点已通过蒙特卡洛模拟得到验证。本研究结果表明，图on上的伊辛模型兼具可精确求解平均场模型的分析可处理性，以及适配广泛网络拓扑的能力，我们期望在自旋模型中使用图on将为复杂网络上相互作用系统的统计物理学带来新见解。
英文摘要：
Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erdős-Rényi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.
