{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T04:01:43Z","timestamp":1649131303025},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2020,10,27]],"date-time":"2020-10-27T00:00:00Z","timestamp":1603756800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is known that for <jats:italic>K<\/jats:italic><jats:sub><jats:italic>n,n<\/jats:italic><\/jats:sub> equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000425_inline1.png\" \/><jats:tex-math>\n$\\zeta(2)=\\pi^2\/6$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in probability. Similar convergence has been established for all edge cost distributions of <jats:italic>pseudo-dimension<\/jats:italic><jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000425_inline2.png\" \/><jats:tex-math>\n$q \\geq 1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In this paper we extend those results to all real positive <jats:italic>q<\/jats:italic>, confirming the M\u00e9zard\u2013Parisi conjecture in the last remaining applicable case.<\/jats:p>","DOI":"10.1017\/s0963548320000425","type":"journal-article","created":{"date-parts":[[2020,10,27]],"date-time":"2020-10-27T09:40:37Z","timestamp":1603791637000},"page":"374-397","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["The minimum perfect matching in pseudo-dimension 0 &lt; <i>q<\/i> &lt; 1"],"prefix":"10.1017","volume":"30","author":[{"given":"Joel","family":"Larsson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2020,10,27]]},"reference":[{"key":"S0963548320000425_ref4","doi-asserted-by":"crossref","unstructured":"[4] Karp, R. M. (1987) An upper bound on the expected cost of an optimal assignment. In Discrete Algorithms and Complexity, Vol. 15 of Perspectives in Computing, pp. 1\u20134. Academic Press.","DOI":"10.1016\/B978-0-12-386870-1.50006-X"},{"key":"S0963548320000425_ref3","doi-asserted-by":"crossref","unstructured":"[3] Aldous, D. and Steele, J. M. (2004) The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures ( Kesten, H. , ed.), Vol. 110 of Encyclopaedia of Mathematical Sciences, pp. 1\u201372. Springer.","DOI":"10.1007\/978-3-662-09444-0_1"},{"key":"S0963548320000425_ref12","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1137\/0208036","article-title":"On the expected value of a random assignment problem","volume":"8","author":"Walkup","year":"1979","journal-title":"SIAM J. 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Fields"},{"key":"S0963548320000425_ref14","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.4007\/annals.2012.175.3.2","article-title":"Replica symmetry of the minimum matching","volume":"175","author":"W\u00e4stlund","year":"2012","journal-title":"Ann. of Math."},{"key":"S0963548320000425_ref8","doi-asserted-by":"crossref","first-page":"L771","DOI":"10.1051\/jphyslet:019850046017077100","article-title":"Replicas and optimization","volume":"46","author":"M\u00e9zard","year":"1985","journal-title":"J. Physique Lett."},{"key":"S0963548320000425_ref1","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/BF01192719","article-title":"Asymptotics in the random assignment problem","volume":"93","author":"Aldous","year":"1992","journal-title":"Probab. Theory Rel. Fields"},{"key":"S0963548320000425_ref7","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/060","volume-title":"Large Networks and Graph Limits","author":"Lov\u00e1sz","year":"2012"},{"key":"S0963548320000425_ref9","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1002\/rsa.20084","article-title":"Proofs of the Parisi and Coppersmith\u2013Sorkin random assignment conjectures","volume":"27","author":"Nair","year":"2005","journal-title":"Random Struct. Algorithms"},{"key":"S0963548320000425_ref11","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1287\/moor.1090.0380","article-title":"Belief propagation: an asymptotically optimal algorithm for the random assignment problem","volume":"34","author":"Salez","year":"2009","journal-title":"Math. Oper. Res."},{"key":"S0963548320000425_ref2","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1002\/rsa.1015","article-title":"The \n\n\n\n$\\zeta(2)$\n\n\nlimit in the random assignment problem","volume":"18","author":"Aldous","year":"2001","journal-title":"Random Struct. 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Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}