Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at the p-order according to its properties at the q-order ( p⩽q ). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

中文翻译：

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高阶相互作用的单纯形路径积分和单纯形重整化群*


现代相变和尺度不变性理论植根于路径积分公式和重整化群（RG）。尽管这些方法在仅具有成对相互作用的简单系统中适用，但它们在具有不可分解的高阶相互作用（即任意单元组之间的相互作用）的复杂系统中效果较差。为了精确表征高阶相互作用系统的普遍性，我们提出了单纯形路径积分和单纯形 RG (SRG) 作为任意高阶和异构相互作用的经典方法的推广。我们首先形式化由高阶相互作用控制的单元的轨迹，以基于高阶传播器定义相应单纯形上的路径积分。然后，我们开发了一种方法来整合动量空间中的短程高阶相互作用，并伴随着对高阶相互作用生成的单纯形结构起作用的粗粒度程序。所提出的 SRG 配备了分而治之的框架，可以处理由于高阶相互作用的稀疏分布而导致的遍历性缺失的问题，并且可以在p - 根据其属性排序q -命令 （ p ⩽ q ）。相关的尺度关系及其推论为区分不同阶的尺度不变、弱尺度不变和尺度相关系统提供了支持。 我们在多阶尺度不变性验证、拓扑不变性发现、组织结构识别和信息瓶颈分析方面验证了我们的理论。这些实验证明了我们的理论在系统简化过程中识别高阶相互作用系统的内在统计和拓扑特性的能力。