{"id":158056,"date":"2008-06-19T00:00:00","date_gmt":"2008-06-19T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/proof-of-the-local-rem-conjecture-for-number-partitioning-ii-growing-energy-scales\/"},"modified":"2018-10-16T20:10:05","modified_gmt":"2018-10-17T03:10:05","slug":"proof-of-the-local-rem-conjecture-for-number-partitioning-ii-growing-energy-scales","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/proof-of-the-local-rem-conjecture-for-number-partitioning-ii-growing-energy-scales\/","title":{"rendered":"Proof of the local REM conjecture for number partitioning II: Growing energy scales"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some fixed probability distribution with density p. In Part I of this work, we established the so-called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as n\u2192\u221e, the suitably rescaled energy spectrum above some fixed scale \u03b1 tends to a Poisson process with density one, and the partitions corresponding to these energies become asymptotically uncorrelated. In this part, we analyze the number partitioning problem for energy scales \u03b1n that grow with n, and show that the local REM conjecture holds as long as n-\u00bc\u03b1n\u21920 and fails if \u03b1n grows like kn\u00bc with k>0.<br \/>\nWe also consider the SK-spin glass model, and show that it has an analogous threshold: the local REM conjecture holds for energies of order o(n), and fails if the energies grow like kn with k > 0.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some fixed probability distribution with density p. In Part I of this work, we established the so-called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as n\u2192\u221e, the suitably rescaled energy spectrum above some fixed scale [&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":"","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":"Stephan Mertens, Chandra 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