Condensation of the Roots of Real Random Polynomials on the Real Axis.
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| Title: | Condensation of the Roots of Real Random Polynomials on the Real Axis. |
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| Authors: | Schehr, Grégory1 schehr@th.u-psud.fr, Majumdar, Satya N.2 |
| Source: | Journal of Statistical Physics. May2009, Vol. 135 Issue 4, p587-598. 12p. 1 Diagram, 1 Graph. |
| Subjects: | Random polynomials, Bose-Einstein condensation, Bosons, Condensation, Gaussian processes |
| Abstract: | We introduce a family of real random polynomials of degree n whose coefficients a k are symmetric independent Gaussian variables with variance $\langle a_{k}^{2}\rangle=e^{-k^{\alpha}}$ , indexed by a real α≥0. We compute exactly the mean number of real roots 〈 N n〉 for large n. As α is varied, one finds three different phases. First, for 0≤ α<1, one finds that $\langle N_{n}\rangle \sim (\frac{2}{\pi})\log{n}$ . For 1< α<2, there is an intermediate phase where 〈 N n〉 grows algebraically with a continuously varying exponent, $\langle N_{n}\rangle \sim \frac{2}{\pi}\sqrt{\frac{\alpha-1}{\alpha}}\,n^{\alpha/2}$ . And finally for α>2, one finds a third phase where 〈 N n〉∼ n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots 〈 N n〉/ n are real. This condensation occurs via a localization of the real roots around the values $\pm \exp [\frac{\alpha}{2}(k+\frac{1}{2})^{\alpha-1}]$ , 1≪ k≤ n. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Statistical Physics is the property of Springer Nature 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 |
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| Header | DbId: egs DbLabel: Engineering Source An: 41779773 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Condensation of the Roots of Real Random Polynomials on the Real Axis. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Schehr%2C+Grégory%22">Schehr, Grégory</searchLink><relatesTo>1</relatesTo><i> schehr@th.u-psud.fr</i><br /><searchLink fieldCode="AR" term="%22Majumdar%2C+Satya+N%2E%22">Majumdar, Satya N.</searchLink><relatesTo>2</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Statistical+Physics%22">Journal of Statistical Physics</searchLink>. May2009, Vol. 135 Issue 4, p587-598. 12p. 1 Diagram, 1 Graph. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Random+polynomials%22">Random polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Bose-Einstein+condensation%22">Bose-Einstein condensation</searchLink><br /><searchLink fieldCode="DE" term="%22Bosons%22">Bosons</searchLink><br /><searchLink fieldCode="DE" term="%22Condensation%22">Condensation</searchLink><br /><searchLink fieldCode="DE" term="%22Gaussian+processes%22">Gaussian processes</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We introduce a family of real random polynomials of degree n whose coefficients a k are symmetric independent Gaussian variables with variance $\langle a_{k}^{2}\rangle=e^{-k^{\alpha}}$ , indexed by a real α≥0. We compute exactly the mean number of real roots 〈 N n〉 for large n. As α is varied, one finds three different phases. First, for 0≤ α<1, one finds that $\langle N_{n}\rangle \sim (\frac{2}{\pi})\log{n}$ . For 1< α<2, there is an intermediate phase where 〈 N n〉 grows algebraically with a continuously varying exponent, $\langle N_{n}\rangle \sim \frac{2}{\pi}\sqrt{\frac{\alpha-1}{\alpha}}\,n^{\alpha/2}$ . And finally for α>2, one finds a third phase where 〈 N n〉∼ n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots 〈 N n〉/ n are real. This condensation occurs via a localization of the real roots around the values $\pm \exp [\frac{\alpha}{2}(k+\frac{1}{2})^{\alpha-1}]$ , 1≪ k≤ n. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Statistical Physics is the property of Springer Nature 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: 10.1007/s10955-009-9755-8 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 12 StartPage: 587 Subjects: – SubjectFull: Random polynomials Type: general – SubjectFull: Bose-Einstein condensation Type: general – SubjectFull: Bosons Type: general – SubjectFull: Condensation Type: general – SubjectFull: Gaussian processes Type: general Titles: – TitleFull: Condensation of the Roots of Real Random Polynomials on the Real Axis. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Schehr, Grégory – PersonEntity: Name: NameFull: Majumdar, Satya N. IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 05 Text: May2009 Type: published Y: 2009 Identifiers: – Type: issn-print Value: 00224715 Numbering: – Type: volume Value: 135 – Type: issue Value: 4 Titles: – TitleFull: Journal of Statistical Physics Type: main |
| ResultId | 1 |
