Counting-Based Effective Dimension and Discrete Regularizations

• URL: http://arxiv.org/abs/2205.11520v1
• Date: Mon, 23 May 2022 17:35:39 GMT
• Title: Counting-Based Effective Dimension and Discrete Regularizations
• Authors: Ivan Horv\'ath, Peter Marko\v{s}, Robert Mendris
• Abstract summary: Effective number theory determines all additive ways to assign counts to collections of objects with probabilities or other additive weights. This effective counting dimension (ECD) specifies how the number of objects in a support scales with their total number. ECD can be used to characterize targets of discrete regularizations in physics and other quantitative sciences.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Effective number theory determines all additive ways to assign counts to collections of objects with probabilities or other additive weights. Here we construct all counting-based schemes to select effective supports on such collections, and show that it leads to a unique notion of effective dimension. This effective counting dimension (ECD) specifies how the number of objects in a support scales with their total number, and its uniqueness means that all schemes yield the same value. Hence, ECD is well defined and can be used to characterize targets of discrete regularizations in physics and other quantitative sciences. Given its generality, ECD may help to connect and interpret results from widely distinct areas. Our analysis makes recent studies of effective spatial dimensions in lattice quantum chromodynamics and Anderson localization models well founded. We address the reliability of regularization removals in practice and perform the respective numerical analysis in the context of 3d Anderson criticality. Our arguments suggest that measure-based dimensions (Minkowski, Hausdorff) of fixed sets have good probabilistic extensions.

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