Counting-Based Effective Dimension and Discrete Regularizations
Status PubMed-not-MEDLINE Language English Country Switzerland Media electronic
Document type Journal Article
Grant support
1/0101/20
Slovak Grant Agency Vega
PubMed
36981369
PubMed Central
PMC10048299
DOI
10.3390/e25030482
PII: e25030482
Knihovny.cz E-resources
- Keywords
- Anderson localization, Minkowski dimension, effective counting dimension, effective description, effective number theory, effective support, lattice QCD, minimal effective description, regularization,
- Publication type
- Journal Article MeSH
Fractal-like structures of varying complexity are common in nature, and measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric characterization. However, at the level of fundamental dynamics, which is quantum, structure does not enter via geometric features of fixed sets but is encoded in probability distributions on associated spaces. The question then arises whether a robust notion of the fractal measure-based dimension exists for structures represented in this way. Starting from effective number theory, we construct all counting-based schemes to select effective supports on collections of objects with probabilities and associate the effective counting dimension (ECD) with each. We then show that the ECD is scheme-independent and, thus, a well-defined measure-based dimension whose meaning is analogous to the Minkowski dimension of fixed sets. In physics language, ECD characterizes probabilistic descriptions arising in a theory or model via discrete "regularization". For example, our analysis makes recent surprising results on effective spatial dimensions in quantum chromodynamics and Anderson models well founded. We discuss how to assess the reliability of regularization removals in practice and perform such analysis in the context of 3d Anderson criticality.
Department of Mathematics Shawnee State University Portsmouth OH 45662 USA
Department of Physics and Astronomy University of Kentucky Lexington KY 40506 USA
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