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[{"id":74438604,"identifier":"Attachment_74438604","shouldShowBulkDownload":false}]; window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":61395143,"created_at":"2021-11-09T04:57:11.520-08:00","from_world_paper_id":183495233,"updated_at":"2025-01-13T05:51:05.620-08:00","_data":{"publisher":"Springer","ai_abstract":"The presented analysis investigates the relationship between the High Temperature Ising Model on a triangular lattice and critical Bernoulli percolation models. By exploring the properties and underlying mechanisms of phase transitions within these models, the research elucidates the critical behavior indicative of percolation thresholds. Furthermore, significant implications for understanding percolation phenomena in statistical mechanics are discussed.","publication_date":"2010,,","publication_name":"Journal of Statistical Physics"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [52848710]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "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":74438604,"attachmentType":"pdf"}"><img alt="First page of “The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/74438604/mini_magick20220707-2057-1k46big.png?1657203640" /><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">The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="52848710" href="https://independent.academia.edu/FedericoCamia"><img alt="Profile image of Federico Camia" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Federico Camia</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2010, Journal of Statistical Physics</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">22 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 = 61395143; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=61395143"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); 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--detail ds2-5-body-md">AI-generated Abstract</p><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">The presented analysis investigates the relationship between the High Temperature Ising Model on a triangular lattice and critical Bernoulli percolation models. By exploring the properties and underlying mechanisms of phase transitions within these models, the research elucidates the critical behavior indicative of percolation thresholds. 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data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="74438604" data-landing_url="https://www.academia.edu/61395143/The_High_Temperature_Ising_Model_on_the_Triangular_Lattice_is_a_Critical_Bernoulli_Percolation_Model" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="89448905" 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/89448905/The_critical_region_of_the_random_bond_Ising_model">The critical region of the random-bond Ising model</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="44218800" href="https://independent.academia.edu/AndreiTalapov">Andrei Talapov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics: Condensed Matter, 1994</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 critical region of the random-bond Ising model","attachmentId":93253234,"attachmentType":"pdf","work_url":"https://www.academia.edu/89448905/The_critical_region_of_the_random_bond_Ising_model","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/89448905/The_critical_region_of_the_random_bond_Ising_model"><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="31806789" 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/31806789/Two_dimensional_XXZ_Ising_model_on_a_square_hexagon_lattice">Two-dimensional XXZ-Ising model on a square-hexagon 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="38763024" href="https://ufla.academia.edu/OnofreRojas">Onofre Rojas</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2009</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":"Two-dimensional XXZ-Ising model on a square-hexagon lattice","attachmentId":52105017,"attachmentType":"pdf","work_url":"https://www.academia.edu/31806789/Two_dimensional_XXZ_Ising_model_on_a_square_hexagon_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/31806789/Two_dimensional_XXZ_Ising_model_on_a_square_hexagon_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="2" data-entity-id="69177743" 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/69177743/The_high_temperature_Ising_model_on_the_triangular_lattice_is_a_critical_percolation_model">The high temperature Ising model on the triangular lattice is a critical percolation model</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">Eprint Arxiv 0806 3020, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">The Ising model at inverse temperature $\beta$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-\beta}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $\beta$, this generates a dependent (spin) percolation model with parameter $r$. We show that, on the triangular lattice and for $\beta<\beta_c$, this model has a percolation phase transition at $r=1/2$, corresponding to the Ising model. This sheds some light on the conjecture that the high temperature Ising model on the triangular lattice is in the percolation universality class and that its scaling limit can be described in terms of SLE$_6$. We also prove uniqueness of the infinite +1 cluster for $r>1/2$, sharpness of the percolation phase transition (by showing exponential decay of the cluster size distribution for $r<1/2$), and continuity of the percolation function for all $r \in [0,1]$.</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 high temperature Ising model on the triangular lattice is a critical percolation model","attachmentId":79372558,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177743/The_high_temperature_Ising_model_on_the_triangular_lattice_is_a_critical_percolation_model","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/69177743/The_high_temperature_Ising_model_on_the_triangular_lattice_is_a_critical_percolation_model"><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="47381175" 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/47381175/Exact_solution_of_Ising_model_on_a_small_world_network">Exact solution of Ising model on a small-world network</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="241397" href="https://up-pt.academia.edu/YuriyPogorelov">Yuriy Pogorelov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2004</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":"Exact solution of Ising model on a small-world network","attachmentId":66502066,"attachmentType":"pdf","work_url":"https://www.academia.edu/47381175/Exact_solution_of_Ising_model_on_a_small_world_network","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/47381175/Exact_solution_of_Ising_model_on_a_small_world_network"><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="29970620" 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/29970620/The_Fuchsian_differential_equation_of_the_square_lattice_Ising_model_3_susceptibility">The Fuchsian differential equation of the square lattice Ising model (3) susceptibility</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="57329585" href="https://independent.academia.edu/SaoudHassani">Saoud Hassani</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="56808212" href="https://independent.academia.edu/NZenine">N. Zenine</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="57058786" href="https://independent.academia.edu/MaillardJm">J-m Maillard</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 2004</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 Fuchsian differential equation of the square lattice Ising model (3) susceptibility","attachmentId":50429742,"attachmentType":"pdf","work_url":"https://www.academia.edu/29970620/The_Fuchsian_differential_equation_of_the_square_lattice_Ising_model_3_susceptibility","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/29970620/The_Fuchsian_differential_equation_of_the_square_lattice_Ising_model_3_susceptibility"><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="33303731" 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/33303731/Monte_Carlo_study_of_the_phase_transition_in_the_critical_behavior_of_the_Ising_model_with_shear">Monte Carlo study of the phase transition in the critical behavior of the Ising model with shear</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="65019075" href="https://independent.academia.edu/GustavoSaracco">Gustavo Saracco</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2009</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":"Monte Carlo study of the phase transition in the critical behavior of the Ising model with shear","attachmentId":53368156,"attachmentType":"pdf","work_url":"https://www.academia.edu/33303731/Monte_Carlo_study_of_the_phase_transition_in_the_critical_behavior_of_the_Ising_model_with_shear","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/33303731/Monte_Carlo_study_of_the_phase_transition_in_the_critical_behavior_of_the_Ising_model_with_shear"><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="28560638" 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/28560638/Phase_transition_in_the_random_triangle_model">Phase transition in the random triangle model</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="53562305" href="https://independent.academia.edu/JohanJonasson2">Johan Jonasson</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Applied Probability, 1999</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":"Phase transition in the random triangle model","attachmentId":48917016,"attachmentType":"pdf","work_url":"https://www.academia.edu/28560638/Phase_transition_in_the_random_triangle_model","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/28560638/Phase_transition_in_the_random_triangle_model"><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="69177767" 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/69177767/The_Percolation_Transition_in_the_Zero_Temperature_Domany_Model">The Percolation Transition in the Zero-Temperature Domany Model</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">2003</p><p class="ds-related-work--abstract ds2-5-body-sm">We analyze a deterministic cellular automaton σ · = (σn: n ≥ 0) corresponding to the zero-temperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice H. The state space SH = {−1,+1} H consists of assignments of −1 or +1 to each site of H and the initial state σ0 = {σ0 x}x∈H is chosen randomly with P(σ0 x = +1) = p ∈ [0,1]. The sites of H are partitioned in two sets A and B so that all the neighbors of a site x in A belong to B and vice versa, and the discrete time dynamics is such that the σ · x’s with x ∈ A (respectively, B) are updated simultaneously at odd (resp., even) times, making σ · x agree with the majority of its three neighbors. In [1] it was proved that there is a percolation transition at p = 1/2 in the percolation models defined by σ n, for all times n ∈ [1, ∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν and η of the dependent percolation models defined by σ n,n ∈ [1, ∞], have the same values ...</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 Percolation Transition in the Zero-Temperature Domany Model","attachmentId":79372674,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177767/The_Percolation_Transition_in_the_Zero_Temperature_Domany_Model","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/69177767/The_Percolation_Transition_in_the_Zero_Temperature_Domany_Model"><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="22106499" 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/22106499/Phase_diagram_and_critical_behavior_of_the_spin_1_Baxter_Wu_model_with_a_crystal_field">Phase diagram and critical behavior of the spin-1 Baxter-Wu model with a crystal field</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="43414141" href="https://independent.academia.edu/JPlascak">J. 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