Condensation in Nongeneric Trees
We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcri...
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| Veröffentlicht in: | Journal of statistical physics 2011-01, Vol.142 (2), p.277-313 |
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| container_title | Journal of statistical physics |
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| creator | Jonsson, Thordur Stefánsson, Sigurdur Örn |
| description | We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability
m |
| doi_str_mv | 10.1007/s10955-010-0104-8 |
| format | Article |
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m
<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1−
m
)
N
where
N
is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2
β
−2 if the weight
w
n
of a vertex of degree
n
is asymptotic to
n
−
β
.</description><identifier>ISSN: 0022-4715</identifier><identifier>EISSN: 1572-9613</identifier><identifier>DOI: 10.1007/s10955-010-0104-8</identifier><language>eng</language><publisher>Boston: Springer US</publisher><subject>Mathematical and Computational Physics ; Physical Chemistry ; Physics ; Physics and Astronomy ; Quantum Physics ; Statistical Physics and Dynamical Systems ; Theoretical ; Thermodynamics</subject><ispartof>Journal of statistical physics, 2011-01, Vol.142 (2), p.277-313</ispartof><rights>Springer Science+Business Media, LLC 2010</rights><rights>COPYRIGHT 2011 Springer</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c327t-5fefd8ef9da3f3f58a2c96348808ac803f85d308647b638618ce787e6a96d523</citedby><cites>FETCH-LOGICAL-c327t-5fefd8ef9da3f3f58a2c96348808ac803f85d308647b638618ce787e6a96d523</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/s10955-010-0104-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10955-010-0104-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,777,781,27905,27906,41469,42538,51300</link.rule.ids></links><search><creatorcontrib>Jonsson, Thordur</creatorcontrib><creatorcontrib>Stefánsson, Sigurdur Örn</creatorcontrib><title>Condensation in Nongeneric Trees</title><title>Journal of statistical physics</title><addtitle>J Stat Phys</addtitle><description>We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability
m
<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1−
m
)
N
where
N
is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2
β
−2 if the weight
w
n
of a vertex of degree
n
is asymptotic to
n
−
β
.</description><subject>Mathematical and Computational Physics</subject><subject>Physical Chemistry</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Statistical Physics and Dynamical Systems</subject><subject>Theoretical</subject><subject>Thermodynamics</subject><issn>0022-4715</issn><issn>1572-9613</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNp9kM1KQzEQhYMoWKsP4O6-QHSS3CSTZSn-QdFN9yHmJiWlTSSpC9_elOtahsPAYb6Bcwi5Z_DAAPRjY2CkpMDgrJHiBVkwqTk1iolLsgDgnI6ayWty09oeAAwauSDDuuQp5OZOqeQh5eG95F3IoSY_bGsI7ZZcRXdo4e5vL8n2-Wm7fqWbj5e39WpDveD6RGUMccIQzeREFFGi494oMSICOo8gIspJAKpRfyqBiqEPGnVQzqhJcrEkD_PbnTsEm3Isp-p8nykcky85xNT9lZCIo9FMd4DNgK-ltRqi_arp6OqPZWDPldi5EtvrOGu02Bk-M63f9pjV7st3zT3WP9AvexpiAg</recordid><startdate>20110101</startdate><enddate>20110101</enddate><creator>Jonsson, Thordur</creator><creator>Stefánsson, Sigurdur Örn</creator><general>Springer US</general><general>Springer</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20110101</creationdate><title>Condensation in Nongeneric Trees</title><author>Jonsson, Thordur ; Stefánsson, Sigurdur Örn</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c327t-5fefd8ef9da3f3f58a2c96348808ac803f85d308647b638618ce787e6a96d523</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Mathematical and Computational Physics</topic><topic>Physical Chemistry</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Statistical Physics and Dynamical Systems</topic><topic>Theoretical</topic><topic>Thermodynamics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Jonsson, Thordur</creatorcontrib><creatorcontrib>Stefánsson, Sigurdur Örn</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of statistical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Jonsson, Thordur</au><au>Stefánsson, Sigurdur Örn</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Condensation in Nongeneric Trees</atitle><jtitle>Journal of statistical physics</jtitle><stitle>J Stat Phys</stitle><date>2011-01-01</date><risdate>2011</risdate><volume>142</volume><issue>2</issue><spage>277</spage><epage>313</epage><pages>277-313</pages><issn>0022-4715</issn><eissn>1572-9613</eissn><abstract>We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability
m
<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1−
m
)
N
where
N
is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2
β
−2 if the weight
w
n
of a vertex of degree
n
is asymptotic to
n
−
β
.</abstract><cop>Boston</cop><pub>Springer US</pub><doi>10.1007/s10955-010-0104-8</doi><tpages>37</tpages></addata></record> |
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| subjects | Mathematical and Computational Physics Physical Chemistry Physics Physics and Astronomy Quantum Physics Statistical Physics and Dynamical Systems Theoretical Thermodynamics |
| title | Condensation in Nongeneric Trees |
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