Abstract The behavior of $$b=2$$ b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical $$q=2,3,4$$ q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice $$L_g$$ L g and allowing the source configuration size $$L_0$$ L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that $$b=2$$ b = 2 weighted majority-like RSRG maps acting on the $$q=2$$ q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.

arXiv:2403.14955Journal of Statistical PhysicsAug 27, 20240 citations

Renormalization Group Analysis

Evgeny Barkhudarov

Springer thesesJan 1, 20141 citation

Springer eBooks

Quantitative Form of the Renormalization Group Mapping

Lothar Schäfer

Published Jan 1, 1999• 0 citations

Publisher

DOI: 10.1007/978-3-642-60093-7_13

Related papers

Renormalization Group Transformations as Symmetry Mappings

J. Lukieeski

De Gruyter eBooksDec 31, 19770 citations

Renormalization Group Transformations as Symmetry Mappings

Jerzy Lukierski B

Fortschritte der PhysikJan 1, 19770 citations

Generalization on renormalization group

อารีฟ บุญเสือ

Mar 25, 20260 citations

Scaling and Renormalization in Neural Networks: An Effective Renormalization-Group Description of Grokking

Min Chen, Junqi Wang, Zeguan Wu +2 more

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.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.

cs.LGCross-listedarXiv:2610.36113Oct 28, 2026

Faithfulness of Real-Space Renormalization Group Maps

Katsuya O. Akamatsu, Naoki Kawashima

Abstract The behavior of $b=2$ b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical $q=2,3,4$ q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice $L_g$ L g and allowing the source configuration size $L_0$ L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that $b=2$ b = 2 weighted majority-like RSRG maps acting on the $q=2$ q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.Abstract The behavior of $$b=2$$ b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical $$q=2,3,4$$ q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice $$L_g$$ L g and allowing the source configuration size $$L_0$$ L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that $$b=2$$ b = 2 weighted majority-like RSRG maps acting on the $$q=2$$ q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.

arXiv:2403.14955Journal of Statistical PhysicsAug 27, 20240 citations

Renormalization Group Analysis

Evgeny Barkhudarov

Springer thesesJan 1, 20141 citation

Abstract The behavior of

b=2

b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical

q=2,3,4

q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice

L_g

L g and allowing the source configuration size

L_0

L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that

b=2

b = 2 weighted majority-like RSRG maps acting on the

q=2

q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.