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<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="LT9h1kBgX9UMqz7yH8gHKjhLfA2edy8r-X-z0azeoF1CTBoQnGiw7VRyiiTryhAdizqaud_9OTMuWFuKWeHJrQ" /> <meta name="citation_title" content="Fat fractal percolation and k-fractal percolation" /> <meta name="citation_journal_title" content="Latin American journal of probability and mathematical statistics" /> <meta name="citation_author" content="Federico Camia" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_fractal_percolation" /> <meta name="twitter:title" content="Fat fractal percolation and k-fractal percolation" /> <meta name="twitter:description" content="We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_fractal_percolation" /> <meta property="og:title" content="Fat fractal percolation and k-fractal percolation" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then" /> <meta property="article:author" content="https://independent.academia.edu/FedericoCamia" /> <meta name="description" content="We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then" /> <title>(PDF) Fat fractal percolation and k-fractal percolation</title> <link rel="canonical" href="https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_fractal_percolation" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = 'dc2ad41da5d7ea682babd20f90650302fb0a3a36'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1739804592000); window.Aedu.timeDifference = new Date().getTime() - 1739804592000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \\prod p_n \u0026amp;amp;gt; 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of co...","author":[{"@context":"https://schema.org","@type":"Person","name":"Federico Camia","url":"https://independent.academia.edu/FedericoCamia"}],"contributor":[],"dateCreated":"2022-01-22","dateModified":"2022-01-22","headline":"Fat fractal percolation and k-fractal percolation","image":"https://attachments.academia-assets.com/79372658/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Applied Mathematics","Statistics"],"publication":"Latin American journal of probability and mathematical statistics","publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}],"thumbnailUrl":"https://attachments.academia-assets.com/79372658/thumbnails/1.jpg","url":"https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_fractal_percolation"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: 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{"work":{"id":69177732,"created_at":"2022-01-22T09:14:30.155-08:00","from_world_paper_id":193032319,"updated_at":"2024-09-03T17:20:01.769-07:00","_data":{"abstract":"We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \\prod p_n \u0026gt; 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of co...","publication_name":"Latin American journal of probability and mathematical statistics"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Fat fractal percolation and k-fractal percolation","broadcastable":true,"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.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</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":79372658,"attachmentType":"pdf"}"><img alt="First page of “Fat fractal percolation and k-fractal percolation”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/79372658/mini_magick20220122-15916-16qzktt.png?1642871796" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.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">Fat fractal percolation and k-fractal percolation</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">Latin American journal of probability and mathematical statistics</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">27 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 = 69177732; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n &gt; 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of co...</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":79372658,"attachmentType":"pdf","workUrl":"https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_fractal_percolation"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":79372658,"attachmentType":"pdf","workUrl":"https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_fractal_percolation"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional "sheets" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that p c (N, 2) converges, as N → ∞, to the critical density p c of site percolation on the square lattice. Assuming the existence of</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":"Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process","attachmentId":79372642,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177704/Large_N_Limit_of_Crossing_Probabilities_Discontinuity_and_Asymptotic_Behavior_of_Threshold_Values_in_Mandelbrots_Fractal_Percolation_Process","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/69177704/Large_N_Limit_of_Crossing_Probabilities_Discontinuity_and_Asymptotic_Behavior_of_Threshold_Values_in_Mandelbrots_Fractal_Percolation_Process"><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="1" data-entity-id="61395145" 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/61395145/Universal_Behavior_of_Connectivity_Properties_in_Fractal_Percolation_Models">Universal Behavior of Connectivity Properties in Fractal Percolation Models</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="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Electronic Journal of Probability, 2010</p><p class="ds-related-work--abstract ds2-5-body-sm">Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d ≥ 2. 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.</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":"Universal Behavior of Connectivity Properties in Fractal Percolation Models","attachmentId":74438601,"attachmentType":"pdf","work_url":"https://www.academia.edu/61395145/Universal_Behavior_of_Connectivity_Properties_in_Fractal_Percolation_Models","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/61395145/Universal_Behavior_of_Connectivity_Properties_in_Fractal_Percolation_Models"><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="2" data-entity-id="118703795" 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/118703795/The_geometry_of_fractal_percolation">The geometry of fractal percolation</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="132755638" href="https://bme.academia.edu/KarolySimon">Karoly Simon</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: • the statements work for all directions, not almost all, • the statements are true for more general projections, for example radial projections onto a circle, • in the case dim H > 1, each projection has not only positive Lebesgue measure but also has nonempty interior.</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":"The geometry of fractal percolation","attachmentId":114268428,"attachmentType":"pdf","work_url":"https://www.academia.edu/118703795/The_geometry_of_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/118703795/The_geometry_of_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="3" 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" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="120805367" 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/120805367/Projections_of_fractal_percolations">Projections of fractal percolations</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="132755638" href="https://bme.academia.edu/KarolySimon">Karoly Simon</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets E ⊂ R 2 which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of E is greater than 1 then the orthogonal projection to every line, the radial projection with every center, and distance set from every point contain intervals.</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":"Projections of fractal percolations","attachmentId":115839671,"attachmentType":"pdf","work_url":"https://www.academia.edu/120805367/Projections_of_fractal_percolations","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/120805367/Projections_of_fractal_percolations"><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="5" data-entity-id="19958827" 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/19958827/Percolation_as_a_fractal_growth_problem">Percolation as a fractal growth problem</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="41222335" href="https://independent.academia.edu/AStella1">A. Stella</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="40828295" href="https://independent.academia.edu/LPietronero">L. Pietronero</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physica A: Statistical Mechanics and its Applications, 1990</p><p class="ds-related-work--abstract ds2-5-body-sm">We consider percolation in two dimensions as a fractal growth problem, and apply to it the theory of fractal growth based on the fixed-scale transformation approach developed for diffusion-limited aggregation and the dielectric breakdown model. This represents an important test for this new theoretical method based on an additional invariance property with respect to the renormalization group. We compute the fractal dimension of the percolating cluster including terms up to third order. The result is D = 1.8830 for the square lattice and D = 1.8650 for the triangular lattice. These values are in excellent agreement with the universal exact result D = 91/48 = 1.8958 and also show the potential of this new method for standard problems.</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 as a fractal growth problem","attachmentId":41239891,"attachmentType":"pdf","work_url":"https://www.academia.edu/19958827/Percolation_as_a_fractal_growth_problem","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/19958827/Percolation_as_a_fractal_growth_problem"><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="6" data-entity-id="32428505" 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/32428505/Percolation_on_a_multifractal_scale_free_planar_stochastic_lattice_and_its_universality_class">Percolation on a multifractal scale-free planar stochastic lattice and its universality class</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="62710135" href="https://independent.academia.edu/kamrulHossain8">Kamrul Hassan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate site percolation on a weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network. Percolation is typically characterized by a threshold value p c at which a transition occurs and by a set of critical exponents β, γ , ν which describe the critical behavior of the percolation probability P (p), mean cluster size S(p), and the correlation length ξ . Besides, the exponent τ characterizes the cluster size distribution function n s (p c ) and the fractal dimension d f characterizes the spanning cluster. We numerically obtain the value of p c and of all the exponents. These results suggest that the percolation on WPSL belong to a separate universality class than on all other planar lattices.</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 scale-free planar stochastic lattice and its universality class","attachmentId":52622835,"attachmentType":"pdf","work_url":"https://www.academia.edu/32428505/Percolation_on_a_multifractal_scale_free_planar_stochastic_lattice_and_its_universality_class","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/32428505/Percolation_on_a_multifractal_scale_free_planar_stochastic_lattice_and_its_universality_class"><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="7" data-entity-id="26525698" 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/26525698/II_Territory_covered_by_N_random_walkers_on_stochastic_fractals_The_percolation_aggregate">II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate</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="50493670" href="https://upv-es.academia.edu/LuisAcedo">Luis Acedo</a></div><p class="ds-related-work--abstract ds2-5-body-sm">The average number SN (t) of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity SN (t) is the result of a double average: an average over random walks on a given lattice followed by an average over different realizations of the lattice. We show for two-dimensional percolation clusters at criticality (and conjecture for other stochastic fractals) that the distribution of the survival probability over these realizations is very broad in Euclidean space but very narrow in the chemical or topological space. This allows us to adapt the formalism developed for Euclidean and deterministic fractal lattices to the chemical language, and an asymptotic series for SN (t) analogous to that found for the non-disordered media is proposed here. The main term is equal to the number of sites (volume) inside a "hypersphere" in the chemical space of radius L[ln(N )/c] 1/v where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast 1 − Γt(ℓ) (the probability that a given site at chemical distance ℓ from the origin is visited by a single random walker by time t) decays for large values of ℓ/L:</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":"II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate","attachmentId":46821742,"attachmentType":"pdf","work_url":"https://www.academia.edu/26525698/II_Territory_covered_by_N_random_walkers_on_stochastic_fractals_The_percolation_aggregate","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/26525698/II_Territory_covered_by_N_random_walkers_on_stochastic_fractals_The_percolation_aggregate"><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="18237064" 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/18237064/On_the_Fractal_dimension_and_correlations_in_percolation_theory">On the Fractal dimension and correlations in percolation theory</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="38221071" href="https://bgu.academia.edu/AmnonAharony">Amnon Aharony</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 1984</p><p class="ds-related-work--abstract ds2-5-body-sm">We discuss the fractal dimension of the infinite cluster at the percolation threshold. Using sealing theory and renormalization group we present an explicit expression for the two-point correlation function within percolation clusters. The fractaI dimension is given by direct integration of this function.</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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="32428506" 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/32428506/New_universality_class_in_percolation_on_multifractal_scale_free_planar_stochastic_lattice">New universality class in percolation on multifractal scale-free planar stochastic lattice</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="62710135" href="https://independent.academia.edu/kamrulHossain8">Kamrul Hassan</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We investigate site percolation on a weighted planar stochastic lattice (WPSL) which is a multifractal and whose dual is a scale-free network. Percolation is typically characterized by percolation threshold pc and by a set of critical exponents β, γ, ν which describe the critical behavior of percolation probability P (p) ∼ (pc − p) β , mean cluster size S ∼ (pc − p) −γ and the correlation length ξ ∼ (pc − p) −ν . Besides, the exponent τ characterizes the cluster size distribution function ns(pc) ∼ s −τ and the fractal dimension d f the spanning cluster. We obtain an exact value for pc and for all these exponents. Our results suggest that the percolation on WPSL belong to a new universality class as its exponents do not share the same value as for all the existing planar lattices.</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 universality class in percolation on multifractal scale-free planar stochastic lattice","attachmentId":52622852,"attachmentType":"pdf","work_url":"https://www.academia.edu/32428506/New_universality_class_in_percolation_on_multifractal_scale_free_planar_stochastic_lattice","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/32428506/New_universality_class_in_percolation_on_multifractal_scale_free_planar_stochastic_lattice"><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 class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":79372658,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":79372658,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_79372658" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="126414245" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/126414245/Fractal_percolation_is_unrectifiable">Fractal percolation is unrectifiable</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="57338246" href="https://independent.academia.edu/Tam%C3%A1sKeleti">Tamás Keleti</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2019</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":"Fractal percolation is unrectifiable","attachmentId":120294225,"attachmentType":"pdf","work_url":"https://www.academia.edu/126414245/Fractal_percolation_is_unrectifiable","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/126414245/Fractal_percolation_is_unrectifiable"><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-related-work-sidebar-card" data-collection-position="1" data-entity-id="49182700" data-sort-order="default"><a class="ds-related-work--title 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href="https://ufrn.academia.edu/GilbertoFalkembachCorso">Gilberto Corso</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Eprint Arxiv Cond Mat 0310779, 2003</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":"Varying critical percolation exponents on a multifractal support","attachmentId":46307576,"attachmentType":"pdf","work_url":"https://www.academia.edu/25953869/Varying_critical_percolation_exponents_on_a_multifractal_support","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/25953869/Varying_critical_percolation_exponents_on_a_multifractal_support"><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-related-work-sidebar-card" data-collection-position="3" data-entity-id="89896103" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/89896103/Physical_Properties_of_a_New_Fractal_Model_of_Percolation_Clusters">Physical Properties of a New Fractal Model of Percolation Clusters</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="31313596" href="https://independent.academia.edu/AAharony">A. Aharony</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 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":"Physical Properties of a New Fractal Model of Percolation Clusters","attachmentId":93609890,"attachmentType":"pdf","work_url":"https://www.academia.edu/89896103/Physical_Properties_of_a_New_Fractal_Model_of_Percolation_Clusters","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/89896103/Physical_Properties_of_a_New_Fractal_Model_of_Percolation_Clusters"><span class="ds2-5-text-link__content">View PDF</span><span 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data-collection-position="5" data-entity-id="88751162" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/88751162/Percolation_on_infinitely_ramified_fractal_networks">Percolation on infinitely ramified fractal networks</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="241020478" href="https://independent.academia.edu/OrlandoSusarreyHuerta">Orlando Susarrey Huerta</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physics Letters A, 2018</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 infinitely ramified fractal 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href="https://www.academia.edu/76741662/Patterns_in_random_fractals">Patterns in random fractals</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="163917498" href="https://independent.academia.edu/VilleSuomala">Ville Suomala</a></div><p class="ds-related-work--metadata ds2-5-body-xs">American Journal of Mathematics, 2020</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":"Patterns in random fractals","attachmentId":84347536,"attachmentType":"pdf","work_url":"https://www.academia.edu/76741662/Patterns_in_random_fractals","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 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Vannimenus</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal de Physique Lettres, 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 intrinsic properties of fractal lattices and percolation clusters","attachmentId":87708010,"attachmentType":"pdf","work_url":"https://www.academia.edu/81783435/On_intrinsic_properties_of_fractal_lattices_and_percolation_clusters","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/81783435/On_intrinsic_properties_of_fractal_lattices_and_percolation_clusters"><span class="ds2-5-text-link__content">View PDF</span><span 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ds2-5-body-link" href="https://www.academia.edu/17152423/Fractal_to_Euclidean_crossover_and_scaling_for_random_walks_on_percolation_clusters_II_Three_dimensional_lattices">Fractal to Euclidean crossover and scaling for random walks on percolation clusters. II. Three-dimensional lattices</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36744155" href="https://independent.academia.edu/PanosArgyrakis">Panos Argyrakis</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32994550" href="https://independent.academia.edu/RaoulKopelman">Raoul Kopelman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Journal of Chemical Physics, 1985</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":"Fractal to Euclidean crossover and scaling for random walks on percolation clusters. II. Three-dimensional lattices","attachmentId":42302303,"attachmentType":"pdf","work_url":"https://www.academia.edu/17152423/Fractal_to_Euclidean_crossover_and_scaling_for_random_walks_on_percolation_clusters_II_Three_dimensional_lattices","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/17152423/Fractal_to_Euclidean_crossover_and_scaling_for_random_walks_on_percolation_clusters_II_Three_dimensional_lattices"><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-related-work-sidebar-card" data-collection-position="16" data-entity-id="54043731" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54043731/Connectivity_properties_of_Mandelbrots_percolation_process">Connectivity properties of Mandelbrot's percolation process</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="42322807" href="https://duke.academia.edu/RDurrett">R. Durrett</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Probability Theory and Related Fields, 1988</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":"Connectivity properties of Mandelbrot's percolation process","attachmentId":70598434,"attachmentType":"pdf","work_url":"https://www.academia.edu/54043731/Connectivity_properties_of_Mandelbrots_percolation_process","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/54043731/Connectivity_properties_of_Mandelbrots_percolation_process"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" 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