论文标题
最大熵随机步行：无限设置和蜘蛛网及其缩放限制的示例
Maximum Entropy Random Walks: the Infinite Setting and the Example of Spider Networks with their Scaling Limits
论文作者
论文摘要
在本文中，我们为无限图上的最大熵随机步行（MERW）建立了固体基础。我们介绍了一个广义定义，该定义扩展了原始概念，以及用于处理这种概括的严格工具。与传统的简单随机步道不同，在本地最大化熵的情况下，沿着路径最大化熵，标志着明显的范式偏移并带来了重大的计算挑战。 Merw最初是由物理学家和计算机科学家引入的，与Parry措施和Doob H-Transforms等概念有联系。我们的方法解决了通过示例和反例说明的存在，独特性和近似的挑战。即使在无限的环境中，Merw也继续最大程度地提高熵率，尽管以较不直接的方式。此外，我们对加权蜘蛛网（包括缩放限制）进行了深入的分析，揭示了无限框架的各种现象特征，尤其是相变。提出了基于平甲类问题的统一规模限制证明。此外，我们考虑了一些扩展模型，其中蜘蛛晶格提供了有价值的见解，突出了研究这些步道以获取一般无限加权图的复杂性。
In this article, we establish solid foundations for the study of Maximal Entropy Random Walks (MERWs) on infinite graphs. We introduce a generalized definition that extends the original concept, along with rigorous tools for handling this generalization. Unlike conventional simple random walks, which maximize entropy locally, MERWs maximize entropy globally along their paths, marking a significant paradigm shift and presenting substantial computational challenges. Originally introduced by physicists and computer scientists in [1], MERWs have connections to concepts such as Parry measures and Doob h-transforms. Our approach addresses the challenges of existence, uniqueness, and approximation, illustrated through examples and counterexamples. Even in the infinite setting, MERWs continue to maximize the entropy rate, albeit in a less direct manner. Additionally, we conduct an in-depth analysis of weighted spider networks, including scaling limits, revealing various phenomena characteristic of the infinite framework, notably a phase transition. A unified proof of scaling limits based on submartingale problems is presented. Furthermore, we consider some extended models, where the spider lattice provides valuable insights, highlighting the complexity of studying these walks for general infinite weighted graphs.
