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We find that in fractal dimensions the percolation threshold" /> <meta name="twitter:image" content="https://0.academia-photos.com/52607579/24747489/23602962/s200_paul.van_der_schoot.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/112261427/Connectedness_percolation_of_fractal_liquids" /> <meta property="og:title" content="Connectedness percolation of fractal liquids" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold" /> <meta property="article:author" content="https://independent.academia.edu/PvanderSchoot" /> <meta name="description" content="We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. 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{"work":{"id":112261427,"created_at":"2023-12-25T04:25:30.634-08:00","from_world_paper_id":246713024,"updated_at":"2025-02-02T03:06:31.063-08:00","_data":{"publisher":"American Physical Society","ai_title_tag":"Percolation in Fractal Liquid Connections","grobid_abstract":"We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold interpolates continuously between integer-dimensional values, and that it decreases monotonically with increasing (fractal) dimension. The influence of hard-core interactions is only significant for dimensions below three. Finally, our theory incorrectly suggests that a percolation threshold is absent below about two dimensions, which we attribute to the breakdown of the connectedness Percus-Yevick closure.","publication_date":"2021,11,10","publication_name":"Physical review","grobid_abstract_attachment_id":"109545464"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Connectedness percolation of fractal liquids","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [52607579]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; 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":109545464,"attachmentType":"pdf"}"><img alt="First page of “Connectedness percolation of fractal liquids”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/109545464/mini_magick20231225-1-rzz2w9.png?1703507207" /><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">Connectedness percolation of fractal liquids</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="52607579" href="https://independent.academia.edu/PvanderSchoot"><img alt="Profile image of Paul van der Schoot" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/52607579/24747489/23602962/s65_paul.van_der_schoot.jpg" />Paul van der Schoot</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2021, Physical review</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">11 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 = 112261427; 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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 apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold interpolates continuously between integer-dimensional values, and that it decreases monotonically with increasing (fractal) dimension. The influence of hard-core interactions is only significant for dimensions below three. Finally, our theory incorrectly suggests that a percolation threshold is absent below about two dimensions, which we attribute to the breakdown of the connectedness Percus-Yevick closure.</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":109545464,"attachmentType":"pdf","workUrl":"https://www.academia.edu/112261427/Connectedness_percolation_of_fractal_liquids"}">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":109545464,"attachmentType":"pdf","workUrl":"https://www.academia.edu/112261427/Connectedness_percolation_of_fractal_liquids"}"><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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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="1" data-entity-id="81783435" 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/81783435/On_intrinsic_properties_of_fractal_lattices_and_percolation_clusters">On intrinsic properties of fractal lattices and percolation clusters</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="48874348" href="https://independent.academia.edu/JVannimenus">J. Vannimenus</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal de Physique Lettres, 1984</p><p class="ds-related-work--abstract ds2-5-body-sm">2014 Certaines propriétés des réseaux fractals sont indépendantes du plongement Euclidien. Les implications de cette invariance sont discutées pour divers problèmes : conductivité d'amas de percolation, inégalités entre dimensions intrinsèques, coefficient de diffusion dans un milieu poreux en présence d'un écoulement. Abstract 2014 Certain properties of fractal lattices are independent of the Euclidean embedding. The implications of this invariance are discussed for various problems : the conductivity of percolation clusters, inequalities between intrinsic dimensions, and the diffusion coefficient in a flow through a porous medium.</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-wsj-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 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="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="3" data-entity-id="89896103" 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/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-wsj-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-wsj-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 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="69177732" 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/69177732/Fat_fractal_percolation_and_k_fractal_percolation">Fat fractal percolation and k-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="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Latin American journal of probability and mathematical statistics</p><p class="ds-related-work--abstract ds2-5-body-sm">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-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":"Fat fractal percolation and k-fractal percolation","attachmentId":79372658,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177732/Fat_fractal_percolation_and_k_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/69177732/Fat_fractal_percolation_and_k_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="5" 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="6" data-entity-id="74007298" 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/74007298/Diffusion_and_fracton_dimensionality_on_fractals_and_on_percolation_clusters">Diffusion and fracton dimensionality on fractals and on percolation clusters</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="36407694" href="https://biu.academia.edu/SHavlin">S. Havlin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1999</p><p class="ds-related-work--abstract ds2-5-body-sm">We present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions. The results for the fractal support the Rammal and Toulouse proposition that d S (N) / d N x B (N) / S (N). The results for percolation are in excellent agreement with the Alexander and Orbach conjecture that the fracton</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":"Diffusion and fracton dimensionality on fractals and on percolation clusters","attachmentId":82317053,"attachmentType":"pdf","work_url":"https://www.academia.edu/74007298/Diffusion_and_fracton_dimensionality_on_fractals_and_on_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/74007298/Diffusion_and_fracton_dimensionality_on_fractals_and_on_percolation_clusters"><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="48351108" 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/48351108/Shortest_path_fractal_dimension_for_percolation_in_two_and_three_dimensions">Shortest-path fractal dimension for percolation in two and three dimensions</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="282899" href="https://umich.academia.edu/RobertZiff">Robert Ziff</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension dmin for percolation in two and three dimensions, using the Leath-Alexandrowicz method which grows a cluster from an active seed site. A variety of quantities are sampled as a function of the chemical distance, including the number of activated sites, a measure of the radius, and the survival probability. By finite-size scaling, we determine dmin = 1.130 77(2) and 1.375 6(6) in two and three dimensions, respectively. The result in two dimensions rules out the recently conjectured value dmin = 217/192 [Phys. Rev. E 81, 020102(R) (2010)].</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":"Shortest-path fractal dimension for percolation in two and three dimensions","attachmentId":67009732,"attachmentType":"pdf","work_url":"https://www.academia.edu/48351108/Shortest_path_fractal_dimension_for_percolation_in_two_and_three_dimensions","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/48351108/Shortest_path_fractal_dimension_for_percolation_in_two_and_three_dimensions"><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="100796622" 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/100796622/Fractals_meet_fractals_self_avoiding_random_walks_on_percolation_clusters">Fractals meet fractals: self-avoiding random walks on percolation clusters</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="67950298" href="https://independent.academia.edu/ViktoriaBlavatska">Viktoria Blavatska</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physics Procedia, 2010</p><p class="ds-related-work--abstract ds2-5-body-sm">The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results yield estimates of critical exponents, governing the scaling laws of disorder averages of the configurational properties of SAWs, and clearly indicate a multifractal spectrum which emerges when two fractals meet each other.</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":"Fractals meet fractals: self-avoiding random walks on percolation clusters","attachmentId":101516306,"attachmentType":"pdf","work_url":"https://www.academia.edu/100796622/Fractals_meet_fractals_self_avoiding_random_walks_on_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/100796622/Fractals_meet_fractals_self_avoiding_random_walks_on_percolation_clusters"><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="5447256" 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/5447256/Fractal_Behavior_of_the_Shortest_Path_Between_Two_Lines_in_Percolation_Systems">Fractal Behavior of the Shortest Path Between Two Lines in Percolation Systems</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="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Using Monte Carlo simulations, we determine the scaling form for the probability distribution of the shortest path l between two lines in a three-dimensional percolation system at criticality; the two lines can have arbitrary positions, orientations, and lengths. We find that the probability distributions can exhibit up to four distinct power-law regimes ͑separated by crossover regimes͒ with exponents depending on the relative orientations of the lines. We explain this rich fractal behavior with scaling arguments.</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 Behavior of the Shortest Path Between Two Lines in Percolation Systems","attachmentId":32570516,"attachmentType":"pdf","work_url":"https://www.academia.edu/5447256/Fractal_Behavior_of_the_Shortest_Path_Between_Two_Lines_in_Percolation_Systems","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/5447256/Fractal_Behavior_of_the_Shortest_Path_Between_Two_Lines_in_Percolation_Systems"><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":109545464,"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":109545464,"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_109545464" 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="5447810" 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/5447810/The_Fractal_Dimension_of_the_Minimum_Path_in_Two_and_Three_Dimensional_Percolation">The Fractal Dimension of the Minimum Path in Two- and Three-Dimensional Percolation</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="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline 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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" 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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/5447809/The_Fractal_Dimension_and_Other_Percolation_Exponents_in_Four_and_Five_Dimensions"><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="6" data-entity-id="27521595" 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/27521595/New_percolating_structures_in_random_fractals_in_2d_and_3d">New percolating structures in random fractals in 2d and 3d</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="51612869" 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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data-author-id="136645528" href="https://independent.academia.edu/DavidMovshovitz">David Movshovitz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 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":"Percolation on infinitely ramified fractals","attachmentId":67567229,"attachmentType":"pdf","work_url":"https://www.academia.edu/49182700/Percolation_on_infinitely_ramified_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 js-related-work-grid-card-view-pdf" href="https://www.academia.edu/49182700/Percolation_on_infinitely_ramified_fractals"><span class="ds2-5-text-link__content">View 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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 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