Simplex path integral and simplex renormalization group for high-order interactions

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Abstract: Modern theories of phase transitions and scale-invariance are rooted in path integral formulation and renormalization groups (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 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 simplexes based on a multi-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. In a special case where all interactions are decomposable, our framework reduces to the well-known Laplacian RG. In more general cases with intertwined high-order interactions, our theory is validated on large-scale data sets of real complex systems (e.g., molecular interactions in proteins and distributed functions in brains) to demonstrate its capacity for identifying intrinsic properties of high-order interacting systems during system reduction.

Has companion code repository: https://github.com/AohuaCheng/Simplex-Renormalization-Group

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