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(PDF) Universal Behavior of Connectivity Properties in Fractal Percolation Models | Federico Camia - Academia.edu
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These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d = 2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter λ. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of λ that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d ≥ 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for d = 2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.","publication_date":"2010,,","publication_name":"Electronic Journal of Probability","grobid_abstract_attachment_id":"74438601"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Universal Behavior of Connectivity Properties in Fractal Percolation Models","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [52848710]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":74438601,"attachmentType":"pdf"}"><img alt="First page of “Universal Behavior of Connectivity Properties in Fractal Percolation Models”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/74438601/mini_magick20211109-23327-p586i6.png?1636462742" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Universal Behavior of Connectivity Properties in Fractal Percolation Models</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="52848710" href="https://independent.academia.edu/FedericoCamia"><img alt="Profile image of Federico Camia" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Federico Camia</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2010, Electronic Journal of Probability</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">29 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 61395145; 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data-collection-position="6" data-entity-id="8100594" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/8100594/Percolation_on_a_multifractal">Percolation on a multifractal</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="15699231" href="https://ucu.academia.edu/EliasdeFreitas">Elias de Freitas</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate percolation phenomena in multifractal objects that are built in a simple way. In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. We observe that the percolation threshold for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent β. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Percolation on a multifractal","attachmentId":48214723,"attachmentType":"pdf","work_url":"https://www.academia.edu/8100594/Percolation_on_a_multifractal","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/8100594/Percolation_on_a_multifractal"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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and Hölder Continuous Curves in Fractal Percolation","attachmentId":74438600,"attachmentType":"pdf","work_url":"https://www.academia.edu/61395147/Dimension_In_equalities_and_H%C3%B6lder_Continuous_Curves_in_Fractal_Percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/61395147/Dimension_In_equalities_and_H%C3%B6lder_Continuous_Curves_in_Fractal_Percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="120805367" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md 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1984</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the Fractal dimension and correlations in percolation theory","attachmentId":39948825,"attachmentType":"pdf","work_url":"https://www.academia.edu/18237064/On_the_Fractal_dimension_and_correlations_in_percolation_theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/18237064/On_the_Fractal_dimension_and_correlations_in_percolation_theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div 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href="https://independent.academia.edu/MichelPerreau">Michel Perreau</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physica A: Statistical Mechanics and its Applications, 1989</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"New percolating structures in random fractals in 2d and 3d","attachmentId":47776883,"attachmentType":"pdf","work_url":"https://www.academia.edu/27521595/New_percolating_structures_in_random_fractals_in_2d_and_3d","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/27521595/New_percolating_structures_in_random_fractals_in_2d_and_3d"><span 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