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

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.