{"id":340112,"date":"2016-12-21T14:15:47","date_gmt":"2016-12-21T22:15:47","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=340112"},"modified":"2018-10-16T21:20:10","modified_gmt":"2018-10-17T04:20:10","slug":"lattice-embeddings-percolation","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/lattice-embeddings-percolation\/","title":{"rendered":"Lattice Embeddings in Percolation"},"content":{"rendered":"<p>Does there exist a Lipschitz injection of Z<em><sup>d<\/sup><\/em> into the open set of a site percolation process on Z<em><sup>D<\/sup><\/em>, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when <em>d<\/em> = <em>D<\/em>, and also when <em>d<\/em> \u2265 2 if the Lipschitz constant <em>M<\/em> is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd, and Scheutzow yields a positive answer for <em>d<\/em> < <em>D<\/em> and <em>M<\/em> = 2. As a result, the above question is answered for all <em>d<\/em>, <em>D<\/em> and <em>M<\/em>. Our proof in the case <em>d<\/em> = <em>D<\/em> uses Tucker\u2019s Lemma from topological combinatorics, together with the aforementioned result for <em>d<\/em> < <em>D<\/em>. One application is an affirmative answer to a question of Peled concerning embeddings of random patterns in two and more dimensions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Does there exist a Lipschitz injection of Zd into the open set of a site percolation process on ZD, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d = D, and also when d \u2265 2 if the Lipschitz constant M is required to be 1. Earlier [&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":"Institute of Mathematical Statistics","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"The Annals of Probability","msr_number":"","msr_organization":"","msr_pages_string":"146-1","msr_page_range_start":"146-1","msr_page_range_end":"","msr_series":"","msr_volume":"40","msr_copyright":"","msr_conference_name":"","msr_doi":"10.1214\/10-AOP615","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":"2010-03-20","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/projecteuclid.org\/euclid.aop\/1325605000","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-340112","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Institute of Mathematical Statistics","msr_edition":"","msr_affiliation":"","msr_published_date":"2010-03-20","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"146-1","msr_chapter":"","msr_isbn":"","msr_journal":"The Annals of Probability","msr_volume":"40","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":"340115","msr_publicationurl":"http:\/\/projecteuclid.org\/euclid.aop\/1325605000","msr_doi":"10.1214\/10-AOP615","msr_publication_uploader":[{"type":"file","title":"embed","viewUrl":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-content\/uploads\/2016\/12\/embed.pdf","id":340115,"label_id":0},{"type":"url","title":"http:\/\/projecteuclid.org\/euclid.aop\/1325605000","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1214\/10-AOP615","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.aop\/1325605000"}],"msr-author-ordering":[{"type":"text","value":"Geoffrey R. 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