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(PDF) Percolation in lattice $k$-neighbor graphs

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We call this graph the directed k-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for k = 1 even the undirected k-neighbor graph never percolates, but the directed one percolates whenever k ≥ d + 1, k ≥ 3 and d ≥ 5, or k ≥ 4 and d = 4. We also show that the undirected 2-neighbor graph percolates for d = 2, the undirected 3-neighbor graph percolates for d = 3, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.","publication_date":"2023,6,26","publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":"112313333"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Percolation in lattice $k$-neighbor graphs","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [49709354]; 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="{&quot;location&quot;:&quot;swp-splash-paper-cover&quot;,&quot;attachmentId&quot;:112313333,&quot;attachmentType&quot;:&quot;pdf&quot;}"><img alt="First page of “Percolation in lattice $k$-neighbor graphs”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/112313333/mini_magick20240311-1-bhmf.png?1710157861" /><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">Percolation in lattice $k$-neighbor graphs</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="49709354" href="https://bme.academia.edu/Andr%C3%A1sT%C3%B3bi%C3%A1s"><img alt="Profile image of András Tóbiás" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/49709354/46281949/35813748/s65_andr_s.t_bi_s.jpg" />András Tóbiás</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2023, arXiv (Cornell University)</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">17 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 = 116085325; 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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 define a random graph obtained via connecting each point of Z d independently to a fixed number 1 ≤ k ≤ 2d of its nearest neighbors via a directed edge. We call this graph the directed k-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for k = 1 even the undirected k-neighbor graph never percolates, but the directed one percolates whenever k ≥ d + 1, k ≥ 3 and d ≥ 5, or k ≥ 4 and d = 4. We also show that the undirected 2-neighbor graph percolates for d = 2, the undirected 3-neighbor graph percolates for d = 3, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;continue-reading-button--work-card&quot;,&quot;attachmentId&quot;:112313333,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;:&quot;https://www.academia.edu/116085325/Percolation_in_lattice_k_neighbor_graphs&quot;}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;download-pdf-button--work-card&quot;,&quot;attachmentId&quot;:112313333,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;:&quot;https://www.academia.edu/116085325/Percolation_in_lattice_k_neighbor_graphs&quot;}"><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-premium-marketing"></div></div><div class="ds-signup-banner ds-signup-banner-premium-marketing"><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="premium-banner-content" data-impression-entity-id="116085325" data-impression-entity-type="2" data-impression-source="premium-banner-desktop"><div class="left"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><span>Get access to the world's latest research</span></div><div class="right"><div class="card free"><div class="header">Free</div><div class="feature-list"><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Download one paper at a time</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Save papers to bookmarks</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Basic search</span></div></div><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--small ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;premium-banner-desktop-free&quot;}">Sign up for free</button></div><div class="card premium"><div class="pill">Recommended</div><div class="header premium">Premium</div><div class="feature-list"><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Get highly curated PDF packages</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Track your impact with Mentions</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Access advanced search filters</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Support Academia’s mission</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Create your personal website</span></div></div><button class="ds2-5-button ds2-5-button--small ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;premium-banner-desktop-upgrade&quot;,&quot;submitText&quot;:&quot;Try Premium for $1&quot;}">Try Premium for $1</button></div></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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1/2$.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Unpredictable paths and percolation&quot;,&quot;attachmentId&quot;:30694333,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/2712846/Unpredictable_paths_and_percolation&quot;,&quot;alternativeTracking&quot;: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/2712846/Unpredictable_paths_and_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="3527451" 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/3527451/Continuum_Percolation_in_the_Relative_Neighborhood_Graph">Continuum Percolation in the Relative Neighborhood Graph</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="3129830" href="https://univ-grenoble-alpes.academia.edu/FranckCorset">Franck Corset</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2010</p><p class="ds-related-work--abstract ds2-5-body-sm">In the present study, we establish the existence of nontrivial site percolation threshold in the Relative Neighborhood Graph (RNG) for Poisson stationary point process with unit intensity in the plane.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Continuum Percolation in the Relative Neighborhood Graph&quot;,&quot;attachmentId&quot;:50246377,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/3527451/Continuum_Percolation_in_the_Relative_Neighborhood_Graph&quot;,&quot;alternativeTracking&quot;: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/3527451/Continuum_Percolation_in_the_Relative_Neighborhood_Graph"><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="120915439" 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/120915439/A_modified_bootstrap_percolation_on_a_random_graph_coupled_with_a_lattice">A modified bootstrap percolation on a random graph coupled with a 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="310634680" href="https://independent.academia.edu/MiklosRuszinko">Miklos Ruszinko</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discrete Applied Mathematics, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper a random graph model G Z 2 N ,p d is introduced, which is a combination of fixed torus grid edges in (Z/N Z) 2 and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u, v ∈ (Z/N Z) 2 with graph distance d on the torus grid is p d = c/N d, where c is some constant. We show that, whp, the diameter D(G Z 2 N ,p d) = Θ(log N). Moreover, we consider a modified non-monotonous bootstrap percolation on G Z 2 N ,p d. We prove the presence of phase transitions in mean-field approximation and provide fairly sharp bounds on the error of the critical parameters.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;A modified bootstrap percolation on a random graph coupled with a lattice&quot;,&quot;attachmentId&quot;:115918876,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/120915439/A_modified_bootstrap_percolation_on_a_random_graph_coupled_with_a_lattice&quot;,&quot;alternativeTracking&quot;: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/120915439/A_modified_bootstrap_percolation_on_a_random_graph_coupled_with_a_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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="19320190" 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/19320190/Density_and_uniqueness_in_percolation">Density and uniqueness in 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="39575601" href="https://uga.academia.edu/RobertBurton">Robert G Burton</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 1989</p><p class="ds-related-work--abstract ds2-5-body-sm">Two results on site percolation on the d-dimensional lattice, d &gt; 1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster has a well-defined density with probability one. The second theorem states that if in addition, the probability measure satisfies the finite energy condition of Newman and Schulman, then there can be at most one infinite cluster with probability one. The simple arguments extend to a broad class of finite-dimensional models, including bond percolation and regular 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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Density and uniqueness in percolation&quot;,&quot;attachmentId&quot;:40554524,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/19320190/Density_and_uniqueness_in_percolation&quot;,&quot;alternativeTracking&quot;: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/19320190/Density_and_uniqueness_in_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="6" data-entity-id="5447510" 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/5447510/Percolation_of_Spatially_Constraint_Networks">Percolation of Spatially Constraint Networks</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">We study how spatial constraints are reflected in the percolation properties of networks embedded in one-dimensional chains and two-dimensional lattices. We assume longrange connections between sites on the lattice where two sites at distance r are chosen to be linked with probability p(r) ∼ r −δ . Similar distributions have been found in spatially embedded real networks such as social and airline networks. We find that for networks embedded in two dimensions, with 2 &lt; δ &lt; 4, the percolation properties show new intermediate behavior different from mean field, with critical exponents that depend on δ. For δ &lt; 2, the percolation transition belongs to the universality class of percolation in Erdös-Rényi networks (mean field), while for δ &gt; 4 it belongs to the universality class of percolation in regular lattices. For networks embedded in one dimension, we find that, for δ &lt; 1, the percolation transition is mean field. For 1 &lt; δ &lt; 2, the critical exponents depend on δ, while for δ &gt; 2 there is no percolation transition as in regular linear chains.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Percolation of Spatially Constraint Networks&quot;,&quot;attachmentId&quot;:32570780,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/5447510/Percolation_of_Spatially_Constraint_Networks&quot;,&quot;alternativeTracking&quot;: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/5447510/Percolation_of_Spatially_Constraint_Networks"><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="120915448" 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/120915448/Bootstrap_percolation_on_a_random_graph_coupled_with_a_lattice">Bootstrap percolation on a random graph coupled with a 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="310634680" href="https://independent.academia.edu/MiklosRuszinko">Miklos Ruszinko</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2016</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper a random graph model G Z 2 N ,p d is introduced, which is a combination of fixed torus grid edges in (Z/N Z) 2 and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u, v ∈ (Z/N Z) 2 with graph distance d on the torus grid is p d = c/N d, where c is some constant. We show that, whp, the diameter D(G Z 2 N ,p d) = Θ(log N). Moreover, we consider non-monotonous bootstrap percolation on G Z 2 N ,p d. We prove the presence of phase transitions in meanfield approximation and provide fairly sharp bounds on the error of the critical parameters. Our model addresses interesting mathematical questions of nonmonotonous bootstrap percolation, and it is motivated by recent results of brain research.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Bootstrap percolation on a random graph coupled with a lattice&quot;,&quot;attachmentId&quot;:115918820,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/120915448/Bootstrap_percolation_on_a_random_graph_coupled_with_a_lattice&quot;,&quot;alternativeTracking&quot;: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/120915448/Bootstrap_percolation_on_a_random_graph_coupled_with_a_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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="3247766" 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/3247766/Directed_percolation_and_generalized_friendly_random_walkers">Directed percolation and generalized friendly random walkers</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="3723086" href="https://cnr-it.academia.edu/francescacolaiori">Francesca Colaiori</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical review letters, 1999</p><p class="ds-related-work--abstract ds2-5-body-sm">We show that the problem of directed percolation on an arbitrary lattice is equivalent to the problem of m directed random walkers with rather general attractive interactions, when suitably continued to m 0. In 1 1 1 dimensions, this is dual to a model of interacting steps on a vicinal surface. A similar correspondence with interacting self-avoiding walks is constructed for isotropic percolation. [S0031-9007(99)08735-9] PACS numbers: 05.40. -a, 05.50. + q, 64.60.Ak</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Directed percolation and generalized friendly random walkers&quot;,&quot;attachmentId&quot;:32245170,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/3247766/Directed_percolation_and_generalized_friendly_random_walkers&quot;,&quot;alternativeTracking&quot;: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/3247766/Directed_percolation_and_generalized_friendly_random_walkers"><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="80112852" 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/80112852/Percolation_type_problems_on_infinite_random_graphs">Percolation-type problems on infinite random graphs</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="18337933" href="https://independent.academia.edu/AlessandroBerarducci">Alessandro Berarducci</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2008</p><p class="ds-related-work--abstract ds2-5-body-sm">We study some percolation problems on the complete graph over N. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory. Contents 1. Introduction 1 2. Notation 5 3. Problem 1 6 4. Extensions and related problems 9 4.1. A notion of capacity for directed graphs 9 4.2. Finite monotone paths and chromatic number 12 5. Problem 2 13 Appendix A. A topological Ramsey theorem 15 Appendix B. A few facts on exchangeable measures 16 References 19</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Percolation-type problems on infinite random graphs&quot;,&quot;attachmentId&quot;:86603437,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/80112852/Percolation_type_problems_on_infinite_random_graphs&quot;,&quot;alternativeTracking&quot;: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/80112852/Percolation_type_problems_on_infinite_random_graphs"><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="{&quot;location&quot;:&quot;continue-reading-button--sticky-ctas&quot;,&quot;attachmentId&quot;:112313333,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;:null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;download-pdf-button--sticky-ctas&quot;,&quot;attachmentId&quot;:112313333,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;: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_112313333" 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. 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href="https://wwu.academia.edu/AmitesSarkar">Amites Sarkar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Advances in Applied Probability, 2005</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Connectivity of random k -nearest-neighbour graphs&quot;,&quot;attachmentId&quot;:46845192,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/11170116/Connectivity_of_random_k_nearest_neighbour_graphs&quot;,&quot;alternativeTracking&quot;: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" 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