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Counting-Based Effective Dimension and Discrete Regularizations

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    0570934 - ÚJF 2024 RIV CH eng J - Článek v odborném periodiku
    Horváth, Ivan - Markoš, P. - Medris, R.
    Counting-Based Effective Dimension and Discrete Regularizations.
    Entropy. Roč. 25, č. 3 (2023), č. článku 482. E-ISSN 1099-4300
    Institucionální podpora: RVO:61389005
    Klíčová slova: Minkowski dimension * effective counting dimension * effective number theory * effective support * effective description * minimal effective description * regularization * Anderson localization * lattice QCD
    Obor OECD: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)
    Impakt faktor: 2.7, rok: 2022
    Způsob publikování: Open access
    https://doi.org/10.3390/e25030482

    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.
    Trvalý link: https://hdl.handle.net/11104/0342273

     
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