Spin Systems on Bethe Lattices
In an extremely influential paper Mézard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random d -regular graph Mézard and Parisi (Eur Phys J B 20:217–233, 2001 ). Their technique was based on certain hy...
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| Veröffentlicht in: | Communications in mathematical physics 2019-12, Vol.372 (2), p.441-523 |
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| description | In an extremely influential paper Mézard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random
d
-regular graph Mézard and Parisi (Eur Phys J B 20:217–233,
2001
). Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases. |
| doi_str_mv | 10.1007/s00220-019-03544-y |
| format | Article |
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d
-regular graph Mézard and Parisi (Eur Phys J B 20:217–233,
2001
). Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases.</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-019-03544-y</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Classical and Quantum Gravitation ; Complex Systems ; Decomposition ; Free energy ; Lattices ; Mathematical and Computational Physics ; Mathematical Physics ; Physics ; Physics and Astronomy ; Quantum Physics ; Relativity Theory ; Theoretical</subject><ispartof>Communications in mathematical physics, 2019-12, Vol.372 (2), p.441-523</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2019</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-362063344211f028c09a92b7bbc0524589559325ae2ed5419cbdc73ba42c88b03</citedby><cites>FETCH-LOGICAL-c319t-362063344211f028c09a92b7bbc0524589559325ae2ed5419cbdc73ba42c88b03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00220-019-03544-y$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00220-019-03544-y$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Coja-Oghlan, Amin</creatorcontrib><creatorcontrib>Perkins, Will</creatorcontrib><title>Spin Systems on Bethe Lattices</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>In an extremely influential paper Mézard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random
d
-regular graph Mézard and Parisi (Eur Phys J B 20:217–233,
2001
). Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases.</description><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Decomposition</subject><subject>Free energy</subject><subject>Lattices</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Theoretical</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAURoMoOI7-ARdScB299-bRZqmDLyi4GF2HNJNqB6etSWfRf2-1gjtXd3O-c-Ewdo5whQD5dQIgAg5oOAglJR8P2AKlIA4G9SFbACBwoVEfs5OUtgBgSOsFu1j3TZutxzSEXcq6NrsNw3vISjcMjQ_plB3V7iOFs9-7ZK_3dy-rR14-PzytbkruBZphEhNoIaQkxBqo8GCcoSqvKg-KpCqMUkaQcoHCRkk0vtr4XFROki-KCsSSXc7ePnaf-5AGu-32sZ1eWhKolBS51hNFM-Vjl1IMte1js3NxtAj2u4OdO9ipg_3pYMdpJOZRmuD2LcQ_9T-rL2hrXWc</recordid><startdate>20191201</startdate><enddate>20191201</enddate><creator>Coja-Oghlan, Amin</creator><creator>Perkins, Will</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20191201</creationdate><title>Spin Systems on Bethe Lattices</title><author>Coja-Oghlan, Amin ; Perkins, Will</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-362063344211f028c09a92b7bbc0524589559325ae2ed5419cbdc73ba42c88b03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Decomposition</topic><topic>Free energy</topic><topic>Lattices</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Coja-Oghlan, Amin</creatorcontrib><creatorcontrib>Perkins, Will</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Coja-Oghlan, Amin</au><au>Perkins, Will</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Spin Systems on Bethe Lattices</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2019-12-01</date><risdate>2019</risdate><volume>372</volume><issue>2</issue><spage>441</spage><epage>523</epage><pages>441-523</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>In an extremely influential paper Mézard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random
d
-regular graph Mézard and Parisi (Eur Phys J B 20:217–233,
2001
). Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-019-03544-y</doi><tpages>83</tpages></addata></record> |
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| subjects | Classical and Quantum Gravitation Complex Systems Decomposition Free energy Lattices Mathematical and Computational Physics Mathematical Physics Physics Physics and Astronomy Quantum Physics Relativity Theory Theoretical |
| title | Spin Systems on Bethe Lattices |
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