Simplex path integral and simplex renormalization group for high-order interactions
Modern theories of phase transitions and scale-invariance are rooted in path integral formulation and renormalization group (RG). Despite the applicability of these approaches on simple systems with only pairwise interactions, they are less effective on complex systems with un-decomposable high-order interactions (i.e., interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose simplex path integral and simplex renormalization group (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 arised from the sparse distribution of high-order interactions and renormalize a system with intertwined high-order interactions on the $p$-order according to its properties on the $q$-order ($p\leq q$). The associated scaling relation and its corollaries support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We have validated our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capacity of our theory for identifying intrinsic statistical and topological properties of high-order interacting systems during system reduction.