Phase transitions for a planar quadratic contact process.
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| Title: | Phase transitions for a planar quadratic contact process. |
|---|---|
| Authors: | Bessonov, Mariya1 mbessonov@citytech.cuny.edu, Durrett, Richard2 rtd@math.duke.edu |
| Source: | Advances in Applied Mathematics. Jun2017, Vol. 87, p82-107. 26p. |
| Subjects: | Phase transitions, Quadratic equations, Algebraic equations, Mathematical models, Symmetric matrices |
| Abstract: | We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values λ f for the process to survive from a finite set and λ e for the existence of a nontrivial stationary distribution. Our first result comes from a standard block construction, while the second involves a comparison with the “generic population model” of Bramson and Gray (1991) [3] . An interesting new feature of our work is the suggestion that, as in the one dimensional contact process, edge speeds characterize critical values. We are able to prove the following for our quadratic contact process when the range is large but suspect they are true for two dimensional finite range attractive particle systems that are symmetric with respect to reflection in each axis. There is a speed c ( θ ) for the expansion of the process in each direction. If c ( θ ) > 0 in all directions, then λ > λ f , while if at least one speed is positive, then λ > λ e . It is a challenging open problem to show that if some speed is negative, then the system dies out from any finite set. [ABSTRACT FROM AUTHOR] |
| Copyright of Advances in Applied Mathematics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 122243066 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Phase transitions for a planar quadratic contact process. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Bessonov%2C+Mariya%22">Bessonov, Mariya</searchLink><relatesTo>1</relatesTo><i> mbessonov@citytech.cuny.edu</i><br /><searchLink fieldCode="AR" term="%22Durrett%2C+Richard%22">Durrett, Richard</searchLink><relatesTo>2</relatesTo><i> rtd@math.duke.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Advances+in+Applied+Mathematics%22">Advances in Applied Mathematics</searchLink>. Jun2017, Vol. 87, p82-107. 26p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Phase+transitions%22">Phase transitions</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+equations%22">Quadratic equations</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+equations%22">Algebraic equations</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink><br /><searchLink fieldCode="DE" term="%22Symmetric+matrices%22">Symmetric matrices</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values λ f for the process to survive from a finite set and λ e for the existence of a nontrivial stationary distribution. Our first result comes from a standard block construction, while the second involves a comparison with the “generic population model” of Bramson and Gray (1991) [3] . An interesting new feature of our work is the suggestion that, as in the one dimensional contact process, edge speeds characterize critical values. We are able to prove the following for our quadratic contact process when the range is large but suspect they are true for two dimensional finite range attractive particle systems that are symmetric with respect to reflection in each axis. There is a speed c ( θ ) for the expansion of the process in each direction. If c ( θ ) > 0 in all directions, then λ > λ f , while if at least one speed is positive, then λ > λ e . It is a challenging open problem to show that if some speed is negative, then the system dies out from any finite set. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Advances in Applied Mathematics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.aam.2017.01.002 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 26 StartPage: 82 Subjects: – SubjectFull: Phase transitions Type: general – SubjectFull: Quadratic equations Type: general – SubjectFull: Algebraic equations Type: general – SubjectFull: Mathematical models Type: general – SubjectFull: Symmetric matrices Type: general Titles: – TitleFull: Phase transitions for a planar quadratic contact process. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Bessonov, Mariya – PersonEntity: Name: NameFull: Durrett, Richard IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2017 Type: published Y: 2017 Identifiers: – Type: issn-print Value: 01968858 Numbering: – Type: volume Value: 87 Titles: – TitleFull: Advances in Applied Mathematics Type: main |
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