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A simplex path integral and a simplex renormalization group for high-order interactions.

Aohua Cheng1,2,3, Yunhui Xu4, Pei Sun5,6

  • 1Department of Psychological and Cognitive Sciences, Tsinghua University, Beijing 100084, People's Republic of China.

Reports on Progress in Physics. Physical Society (Great Britain)
|July 30, 2024
PubMed
Summary
This summary is machine-generated.

We introduce simplex renormalization groups (SRGs) to analyze complex systems with high-order interactions. This new method generalizes path integrals and renormalization groups for better characterization of universality in intricate systems.

Keywords:
complex networkshigh-order interactionsscale invariancesimplex renormalization groupsimplicial complexstatistical physicstopology

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Area of Science:

  • Complex Systems Science
  • Statistical Physics
  • Network Theory

Background:

  • Path integrals and renormalization groups (RGs) are foundational for phase transitions and scale invariance.
  • Classic RG methods are limited in complex systems with high-order, undecomposable interactions.

Purpose of the Study:

  • To generalize path integral formulation and RG methods for systems with arbitrary high-order and heterogeneous interactions.
  • To develop a framework for characterizing universality in complex systems with high-order interactions.

Main Methods:

  • Formalization of unit trajectories under high-order interactions.
  • Development of simplex path integrals and simplex RG (SRG) using high-order propagators.
  • Momentum-space integration of short-range interactions and coarse-graining on simplex structures.
  • Implementation of a divide-and-conquer framework within SRG to handle non-ergodicity and inter-order renormalization (p <= q).

Main Results:

  • The proposed SRG effectively analyzes systems with high-order interactions, overcoming limitations of classic RG.
  • Associated scaling relations differentiate between scale-invariant, weakly scale-invariant, and scale-dependent systems.
  • Validation across diverse applications including scale-invariance verification, topological discovery, and information bottleneck analysis.

Conclusions:

  • The simplex RG provides a robust theoretical tool for understanding complex systems with high-order interactions.
  • The framework accurately identifies intrinsic statistical and topological properties during system reduction.
  • This generalization enhances the analysis of universality across various scientific domains.