{"id":339671,"date":"2016-12-20T14:19:33","date_gmt":"2016-12-20T22:19:33","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&p=339671"},"modified":"2018-10-16T21:15:45","modified_gmt":"2018-10-17T04:15:45","slug":"percolation-games-probabilistic-cellular-automata-hard-core-model","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/percolation-games-probabilistic-cellular-automata-hard-core-model\/","title":{"rendered":"Percolation Games, Probabilistic Cellular Automata, and the Hard-Core Model"},"content":{"rendered":"

Let each site of the square lattice Z2<\/sup> be independently declared closed with probability p<\/em>, and otherwise open<\/em>. Consider the following game: a token starts at the origin, and the two players take turns to move it from its current site x<\/em> to an open site in {x<\/em> + (0, 1), x<\/em> + (1, 0)}; if both these sites are closed, then the player to move loses the game. Is there positive probability that the game is drawn<\/em> with best play \u2013 i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain elementary one-dimensional probabilistic cellular automaton (PCA), which has been studied in the contexts of enumeration of directed animals, the golden-mean subshift, and the hard-core model. Ergodicity of the PCA has been noted as an open problem by several authors. We prove that the PCA is ergodic for all 0 < p<\/em> < 1, and correspondingly that the game on Z2<\/sup> has no draws. We establish similar results for a certain mis\u00e8re variant of the game and a PCA associated to it.<\/p>\n

On the other hand, we prove that the analogous game does exhibit draws for sufficiently small p<\/em> on various directed graphs in higher dimensions, including an oriented version of the even sublattice of Zd<\/em><\/sup> in all d<\/em> \u2265 3. This is proved via a dimension reduction to a hard-core lattice gas in dimension d<\/em>\u22121. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice Zd<\/em><\/sup> for d<\/em> \u2265 3, but here our method encounters a fundamental obstacle.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let each site of the square lattice Z2 be independently declared closed with probability p, and otherwise open. Consider the following game: a token starts at the origin, and the two players take turns to move it from its current site x to an open site in {x + (0, 1), x + (1, 0)}; […]<\/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":"Cornell University Library","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","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":"2015-10-19","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1503.05614","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-339671","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cornell University Library","msr_edition":"","msr_affiliation":"","msr_published_date":"2015-10-19","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","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":"339677","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1503.05614","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"game-dir","viewUrl":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-content\/uploads\/2016\/12\/game-dir.pdf","id":339677,"label_id":0},{"type":"url","title":"https:\/\/arxiv.org\/abs\/1503.05614","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":"https:\/\/arxiv.org\/abs\/1503.05614"}],"msr-author-ordering":[{"type":"user_nicename","value":"holroyd","user_id":32020,"rest_url":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=holroyd"},{"type":"text","value":"Ir\u00e8ne Marcovici","user_id":0,"rest_url":false},{"type":"text","value":"James B. 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