{"id":339884,"date":"2016-12-21T10:03:06","date_gmt":"2016-12-21T18:03:06","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=339884"},"modified":"2018-10-16T21:17:52","modified_gmt":"2018-10-17T04:17:52","slug":"percolation-finite-clusters-infinite-surfaces","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/percolation-finite-clusters-infinite-surfaces\/","title":{"rendered":"Percolation of Finite Clusters and Infinite Surfaces"},"content":{"rendered":"<p>Two related issues are explored for bond percolation on Z<em><sup>d<\/sup><\/em> (with <em>d<\/em> \u2265 3) and its dual plaquette process. Firstly, for what values of the parameter <em>p<\/em> does the complement of the infinite open cluster possess an infinite component? The corresponding critical point <em>p<\/em><sub>fin<\/sub> satisfies <em>p<\/em><sub>fin<\/sub> \u2265 <em>p<\/em><sub>c<\/sub>, and strict inequality is proved when either d is sufficiently large, or <em>d<\/em> \u2265 7 and the model is sufficiently spread out. It is not known whether <em>d<\/em> \u2265 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point <em>p<\/em><sub>surf<\/sub> satisfies <em>p<\/em><sub>surf<\/sub> \u2265 <em>p<\/em><sub>fin<\/sub>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two related issues are explored for bond percolation on Zd (with d \u2265 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin \u2265 pc, and strict inequality is proved when either [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cambridge Philosophical Society","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Mathematical Proceedings of the Cambridge Philosophical Society","msr_number":"","msr_organization":"","msr_pages_string":"263-279","msr_page_range_start":"263","msr_page_range_end":"279","msr_series":"","msr_volume":"156","msr_copyright":"","msr_conference_name":"","msr_doi":"10.1017\/S030500411300073X","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2014-03-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-339884","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cambridge Philosophical Society","msr_edition":"","msr_affiliation":"","msr_published_date":"2014-03-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"263-279","msr_chapter":"","msr_isbn":"","msr_journal":"Mathematical Proceedings of the Cambridge Philosophical Society","msr_volume":"156","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"339887","msr_publicationurl":"","msr_doi":"10.1017\/S030500411300073X","msr_publication_uploader":[{"type":"file","title":"plaq","viewUrl":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-content\/uploads\/2016\/12\/plaq.pdf","id":339887,"label_id":0},{"type":"doi","title":"10.1017\/S030500411300073X","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[],"msr-author-ordering":[{"type":"text","value":"Geoffrey R. 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