The Branching Random Walk and Contact Process on Galton-Watson and Nonhomogeneous Trees
We show that the branching random walk on a Galton-Watson tree may have one or two phase transitions, depending on the relative sizes of the mean degree and the maximum degree. We show that there are some Galton-Watson trees on which the branching random walk has one phase transition while the conta...
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| Veröffentlicht in: | The Annals of probability 2001-10, Vol.29 (4), p.1563-1590 |
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| container_end_page | 1590 |
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| container_title | The Annals of probability |
| container_volume | 29 |
| creator | Pemantle, Robin Stacey, Alan M. |
| description | We show that the branching random walk on a Galton-Watson tree may have one or two phase transitions, depending on the relative sizes of the mean degree and the maximum degree. We show that there are some Galton-Watson trees on which the branching random walk has one phase transition while the contact process has two; this contradicts a conjecture of Madras and Schinazi. We show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree, contradicting a conjecture of Pemantle. |
| doi_str_mv | 10.1214/aop/1015345762 |
| format | Article |
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We show that there are some Galton-Watson trees on which the branching random walk has one phase transition while the contact process has two; this contradicts a conjecture of Madras and Schinazi. We show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree, contradicting a conjecture of Pemantle.</description><identifier>ISSN: 0091-1798</identifier><identifier>EISSN: 2168-894X</identifier><identifier>DOI: 10.1214/aop/1015345762</identifier><identifier>CODEN: APBYAE</identifier><language>eng</language><publisher>Hayward, CA: Institute of Mathematical Statistics</publisher><subject>60K35 ; branching random walk ; Children ; Conditional probabilities ; contact process ; Critical values ; Exact sciences and technology ; Grants ; Infections ; Mathematics ; phase transition ; Plant roots ; Poisson process ; Probability and statistics ; Probability theory and stochastic processes ; Random walk ; Sciences and techniques of general use ; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) ; spectral radius ; Trails ; Tree ; Vertices</subject><ispartof>The Annals of probability, 2001-10, Vol.29 (4), p.1563-1590</ispartof><rights>Copyright 2001 Institute of Mathematical Statistics</rights><rights>2003 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c385t-35bbd6770a4739f3a8a1c83602f4b37b7f34d10716fa3b0bf216a1635b5e1b863</citedby><cites>FETCH-LOGICAL-c385t-35bbd6770a4739f3a8a1c83602f4b37b7f34d10716fa3b0bf216a1635b5e1b863</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2691970$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2691970$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,803,832,885,926,27924,27925,58017,58021,58250,58254</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=13530026$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Pemantle, Robin</creatorcontrib><creatorcontrib>Stacey, Alan M.</creatorcontrib><title>The Branching Random Walk and Contact Process on Galton-Watson and Nonhomogeneous Trees</title><title>The Annals of probability</title><description>We show that the branching random walk on a Galton-Watson tree may have one or two phase transitions, depending on the relative sizes of the mean degree and the maximum degree. 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We show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree, contradicting a conjecture of Pemantle.</description><subject>60K35</subject><subject>branching random walk</subject><subject>Children</subject><subject>Conditional probabilities</subject><subject>contact process</subject><subject>Critical values</subject><subject>Exact sciences and technology</subject><subject>Grants</subject><subject>Infections</subject><subject>Mathematics</subject><subject>phase transition</subject><subject>Plant roots</subject><subject>Poisson process</subject><subject>Probability and statistics</subject><subject>Probability theory and stochastic processes</subject><subject>Random walk</subject><subject>Sciences and techniques of general use</subject><subject>Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)</subject><subject>spectral radius</subject><subject>Trails</subject><subject>Tree</subject><subject>Vertices</subject><issn>0091-1798</issn><issn>2168-894X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2001</creationdate><recordtype>article</recordtype><recordid>eNplUD1PwzAUtBBIlMLKxOCFMa0dJ3ayAREUpAoQSlW26MWx25TUrux04N-TqFE7ML3Pu3fvELqlZEJDGk3B7qaU0JhFseDhGRqFlCdBkkbf52hESEoDKtLkEl15vyGEcCGiEVrma4WfHBi5rs0Kf4Gp7BYvofnBXYoza1qQLf50VirvsTV4Bk1rTbCE1ndVv_Ruzdpu7UoZZfce504pf40uNDRe3QxxjBYvz3n2Gsw_Zm_Z4zyQLInbgMVlWXVKCESCpZpBAlQmjJNQRyUTpdAsqigRlGtgJSl19xNQ3sFiRcuEszF6OPDunN0o2aq9bOqq2Ll6C-63sFAX2WI-dIfQGVWcjOooJgcK6az3TukjmpKid_Y_4H64CV5Co3v3an9CsZgREvba7g57G99ad5yHPKWpIOwP-ZuC6Q</recordid><startdate>20011001</startdate><enddate>20011001</enddate><creator>Pemantle, Robin</creator><creator>Stacey, Alan M.</creator><general>Institute of Mathematical Statistics</general><general>The Institute of Mathematical Statistics</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20011001</creationdate><title>The Branching Random Walk and Contact Process on Galton-Watson and Nonhomogeneous Trees</title><author>Pemantle, Robin ; Stacey, Alan M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c385t-35bbd6770a4739f3a8a1c83602f4b37b7f34d10716fa3b0bf216a1635b5e1b863</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2001</creationdate><topic>60K35</topic><topic>branching random walk</topic><topic>Children</topic><topic>Conditional probabilities</topic><topic>contact process</topic><topic>Critical values</topic><topic>Exact sciences and technology</topic><topic>Grants</topic><topic>Infections</topic><topic>Mathematics</topic><topic>phase transition</topic><topic>Plant roots</topic><topic>Poisson process</topic><topic>Probability and statistics</topic><topic>Probability theory and stochastic processes</topic><topic>Random walk</topic><topic>Sciences and techniques of general use</topic><topic>Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)</topic><topic>spectral radius</topic><topic>Trails</topic><topic>Tree</topic><topic>Vertices</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Pemantle, Robin</creatorcontrib><creatorcontrib>Stacey, Alan M.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><jtitle>The Annals of probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Pemantle, Robin</au><au>Stacey, Alan M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Branching Random Walk and Contact Process on Galton-Watson and Nonhomogeneous Trees</atitle><jtitle>The Annals of probability</jtitle><date>2001-10-01</date><risdate>2001</risdate><volume>29</volume><issue>4</issue><spage>1563</spage><epage>1590</epage><pages>1563-1590</pages><issn>0091-1798</issn><eissn>2168-894X</eissn><coden>APBYAE</coden><abstract>We show that the branching random walk on a Galton-Watson tree may have one or two phase transitions, depending on the relative sizes of the mean degree and the maximum degree. We show that there are some Galton-Watson trees on which the branching random walk has one phase transition while the contact process has two; this contradicts a conjecture of Madras and Schinazi. We show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree, contradicting a conjecture of Pemantle.</abstract><cop>Hayward, CA</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/aop/1015345762</doi><tpages>28</tpages><oa>free_for_read</oa></addata></record> |
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| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; Project Euclid Complete |
| subjects | 60K35 branching random walk Children Conditional probabilities contact process Critical values Exact sciences and technology Grants Infections Mathematics phase transition Plant roots Poisson process Probability and statistics Probability theory and stochastic processes Random walk Sciences and techniques of general use Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) spectral radius Trails Tree Vertices |
| title | The Branching Random Walk and Contact Process on Galton-Watson and Nonhomogeneous Trees |
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