{"id":340106,"date":"2016-12-21T14:10:08","date_gmt":"2016-12-21T22:10:08","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=340106"},"modified":"2018-10-16T21:19:58","modified_gmt":"2018-10-17T04:19:58","slug":"geometry-lipschitz-percolation","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/geometry-lipschitz-percolation\/","title":{"rendered":"Geometry of Lipschitz Percolation"},"content":{"rendered":"<p>We prove several facts concerning Lipschitz percolation, including the following. The critical probability pL for the existence of an open Lipschitz surface in site percolation on Z<em><sup>d<\/sup><\/em> with <em>d<\/em> \u2265 2 satisfies the improved bound pL \u2264 1\u22121\/[8(<em>d<\/em>\u22121)]. Whenever <em>p<\/em> > <em>p<\/em><sub>L<\/sub>, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. For p sufficiently close to 1, the connected regions of Z<em><sup>d\u22121<\/sup><\/em> above which the surface has height 2 or more exhibit stretched-exponential tail behaviour. The last statement is proved via a stochastic inequality stating that the lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z<em><sup>d<\/sup><\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We prove several facts concerning Lipschitz percolation, including the following. The critical probability pL for the existence of an open Lipschitz surface in site percolation on Zd with d \u2265 2 satisfies the improved bound pL \u2264 1\u22121\/[8(d\u22121)]. Whenever p > pL, the height of the lowest Lipschitz surface above the origin has an exponentially [&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":"Institut Henri Poincar\u00c3\u00a9","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Annales de l'Institut Henri Poincar\u00c3\u00a9, Probabilit\u00c3\u00a9s et Statistiques","msr_number":"","msr_organization":"","msr_pages_string":"309-326","msr_page_range_start":"309","msr_page_range_end":"326","msr_series":"","msr_volume":"48","msr_copyright":"","msr_conference_name":"","msr_doi":"10.1214\/10-AIHP403","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":"2010-07-21","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/projecteuclid.org\/euclid.aihp\/1334148200","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-340106","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Institut Henri Poincar\u00c3\u00a9","msr_edition":"","msr_affiliation":"","msr_published_date":"2010-07-21","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"309-326","msr_chapter":"","msr_isbn":"","msr_journal":"Annales de l'Institut Henri Poincar\u00c3\u00a9, Probabilit\u00c3\u00a9s et Statistiques","msr_volume":"48","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":"340109","msr_publicationurl":"http:\/\/projecteuclid.org\/euclid.aihp\/1334148200","msr_doi":"10.1214\/10-AIHP403","msr_publication_uploader":[{"type":"file","title":"hill","viewUrl":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-content\/uploads\/2016\/12\/hill.pdf","id":340109,"label_id":0},{"type":"url","title":"http:\/\/projecteuclid.org\/euclid.aihp\/1334148200","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1214\/10-AIHP403","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":[{"id":0,"url":"http:\/\/projecteuclid.org\/euclid.aihp\/1334148200"}],"msr-author-ordering":[{"type":"user_nicename","value":"holroyd","user_id":32020,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=holroyd"},{"type":"text","value":"G. 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