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arXiv:2401.15899

Fast renormalizing the structures and dynamics of ultra-large systems via random renormalization group

Yang Tian, Yizhou Xu, Pei Sun

Published Jan 29, 2024• 0 citations
arXivHTMLPDF

Abstract

Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.

DOI
10.48550/arxiv.2401.15899

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cs.LGCross-listedarXiv:2610.36113•Oct 28, 2026
Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.
Abstract Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.
Exploring and understanding topological phases in systems with strong distributed disorder requires developing fundamentally new approaches to replace traditional tools such as topological band theory. Here, we present a general real-space renormalization group (RG) approach for scattering models, which is capable of dealing with strong distributed disorder without relying on the renormalization of Hamiltonians or wave functions. Such scheme, based on a block-scattering transformation combined with a replica strategy, is applied for a comprehensive study of strongly disordered unitary scattering networks with localized bulk states, uncovering a connection between topological physics and critical behavior. Our RG scheme leads to topological flow diagrams that unveil how the microscopic competition between reflection and non-reciprocity leads to the large-scale emergence of macroscopic scattering attractors, corresponding to trivial and topological insulators. Our findings are confirmed by a scaling analysis of the localization length (LL) and critical exponents, and experimentally validated. The results not only shed light on the fundamental understanding of topological phase transitions and scaling properties in strongly disordered regimes, but also pave the way for practical applications in modern topological condensed-matter and photonics, where disorder may be seen as a useful design degree of freedom, and no longer as a hindrance.
Far from equilibrium, neural systems self-organize across multiple scales. Exploiting multiscale self-organization in neuroscience and artificial intelligence requires a computational framework for modeling the effective non-equilibrium dynamics of stochastic neural trajectories. Non-equilibrium thermodynamics and representational geometry offer theoretical foundations, but we need scalable data-driven techniques for modeling collective properties of high-dimensional neural networks from partial subsampled observations. Renormalization is a coarse-graining technique central to studying emergent scaling properties of many-body and nonlinear dynamical systems. While widely applied in physics and machine learning, coarse-graining complex dynamical networks remains unsolved, affecting many computational sciences. Recent diffusion-based renormalization, inspired by quantum statistical mechanics, coarse-grains networks near entropy transitions marked by maximal changes in specific heat or information transmission. Here I explore diffusion-based renormalization of neural systems by generating symmetry-breaking representations across scales and offering scalable algorithms using tensor networks. Diffusion-guided renormalization bridges microscale and mesoscale dynamics of dissipative neural systems. For microscales, I developed a scalable graph inference algorithm for discovering community structure from subsampled neural activity. Using community-based node orderings, diffusion-guided renormalization generates renormalization group flow through metagraphs and joint probability functions. Towards mesoscales, diffusion-guided renormalization targets learning the effective non-equilibrium dynamics of dissipative neural trajectories occupying lower-dimensional subspaces, enabling coarse-to-fine control in systems neuroscience and artificial intelligence.
Numerical simulations of critical lattice systems are fundamentally limited by critical slowing down, as long-range correlations are typically established through slow temporal equilibration. A physically constrained generative framework that replaces temporal relaxation with a spatial projection mechanism for critical systems is proposed. Using the two-dimensional Ising model at criticality as a benchmark, we introduce an energy-constrained kernel that synthesizes large-scale configurations from compact equilibrated seeds by enforcing Hamiltonian-level observables. The generated configurations are projected onto the nearest-neighbor energy manifold, ensuring thermodynamic consistency while retaining universal critical properties. We show that the resulting configurations reproduce scale-invariant spin correlations, Binder cumulants, and isotropic structure factors for lattice sizes exceeding 10,000, without iterative Monte Carlo equilibration. While not a strict renormalization group transformation, and motivated by renormalization ideas, the method provides a practical inverse mapping that retains universal features of criticality and enables efficient GPU-parallel generation of ultra-large critical ensembles.
Abstract 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.
Grokking is a delayed transition from memorization to generalization in deep neural networks (NNs). We examine whether scaling and renormalization group (RG) theories can describe this dynamics, establishing a quantitative physical framework for neural generalization transitions. For modular addition on a two-layer NN, an ensemble of trained models defines an order parameter (the squared weight norm) and fluctuation proxies for the heat capacity and the susceptibility, with training epoch and weight decay serving as thermal and field control coordinates. To bridge neural architectures to real-space RG, we construct an exact coupling matrix via a second-order expansion of the loss with respect to per-neuron norms and use its Fiedler vector to define an effective geometry. This spectral ordering yields a monotonic spatial decay of coupling strength that is absent under random permutations. We then execute real-space RG by block-spin aggregating adjacent neurons and parameterizing the coupling matrix into off-diagonal band strengths. Linearizing this local RG map yields relevant scaling exponents and five derived critical exponents, with direct empirical estimates matching RG predictions within 1-10%. Overall, our framework provides a consistent effective description of grokking, suggesting the scaling and renormalization group formalism as a tool to investigate NN dynamics.