Spectral Universality Classes in One-Dimensional Fractal Geometries: From Hierarchical Poisson Limits to Rigid Non-RMT Fractal Spectra

Spectral Universality Classes in One-Dimensional Fractal Geometries: From Hierarchical Poisson Limits to Rigid Non-RMT Fractal Spectra | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 20 March 2026 V1 Latest version Share on Spectral Universality Classes in One-Dimensional Fractal Geometries: From Hierarchical Poisson Limits to Rigid Non-RMT Fractal Spectra Author : Enrique Vidal Silvente 0009-0007-8291-1762 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.177403155.55928694/v1 126 views 53 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We investigate the spectral properties of fractional Laplacians H = L 0.9 defined on several one-dimensional fractal point sets, including Cantor-type hierarchical geometries and subdivision-based fractals such as the Sierpiński and Koch constructions. Using a unified numerical pipeline combining polynomial unfolding, bootstrap-resolved gap statistics, spectral dimension estimation, Brody index analysis, gap-ratio universality, number variance, inverse participation ratios, Rényi entropies, multifractal dimensions, and robustness-to-noise tests, we identify two sharply distinct spectral universality classes. Hierarchical fractals (Cantor and Random Cantor) converge to a Poisson-type spectral regime, with spectral dimension d s ≈ 2, vanishing Brody index β → 0, and gap ratio ⟨r⟩ approaching the Poisson limit. Subdivision fractals (Sierpiński and Koch) exhibit instead a rigid, non-RMT spectral class characterized by extreme level repulsion (⟨r⟩ → 1), Brody index β → 1, low number variance, and multifractal gap statistics, while remaining incompatible with both Poisson and Wigner-Dyson distributions. Scaling across refinement levels (3 ≤ L ≤ 7) reveals stable convergence of all indicators, demonstrating that the linear/integrable Poisson regime is contained as a limiting case within the broader class of hierarchical fractal spectra, whereas subdivision fractals generate a genuinely new universality class with no analogue in classical random matrix theory. These results provide the first systematic classification of spectral universality in one-dimensional fractal geometries and establish a quantitative bridge between geometry, spectral dimension, and level statistics in non-smooth metric spaces. Supplementary Material File (fractal_linear-2.pdf) Download 241.26 KB Information & Authors Information Version history V1 Version 1 20 March 2026 Copyright This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License Keywords cantor fractal linear logical spectra Authors Affiliations Enrique Vidal Silvente 0009-0007-8291-1762 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 126 views 53 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Enrique Vidal Silvente. Spectral Universality Classes in One-Dimensional Fractal Geometries: From Hierarchical Poisson Limits to Rigid Non-RMT Fractal Spectra. Authorea . 20 March 2026. DOI: https://doi.org/10.22541/au.177403155.55928694/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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