Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation.
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| Title: | Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation. |
|---|---|
| Authors: | Schehr, Grégory1 schehr@th.u-psud.fr, Majumdar, Satya N.2 |
| Source: | Journal of Statistical Physics. Jul2008, Vol. 132 Issue 2, p235-273. 39p. 1 Chart, 9 Graphs. |
| Subjects: | Random polynomials, Heat equation, Free probability theory, Quantum chaos, Mean field theory, Rings of integers |
| Abstract: | We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n − θ( d) where θ( d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n −2( θ(2)+ θ( d)). For Weyl polynomials and Binomial polynomials, this probability decays respectively like $\exp{(-2\theta_{\infty}}\sqrt{n})$ and $\exp{(-\pi\theta _{\infty}\sqrt{n})}$ where θ ∞ is such that $\theta(d)=2^{-3/2}\theta_{\infty}\sqrt{d}$ in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [ a, b] has a scaling form given by $\exp{(-N_{ab}\tilde{\varphi}(k/N_{ab}))}$ where N ab is the mean number of real roots in [ a, b] and $\tilde{\varphi}(x)$ a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, ). [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.) | |
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| Items | – Name: Title Label: Title Group: Ti Data: Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation. – 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>. Jul2008, Vol. 132 Issue 2, p235-273. 39p. 1 Chart, 9 Graphs. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Random+polynomials%22">Random polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Heat+equation%22">Heat equation</searchLink><br /><searchLink fieldCode="DE" term="%22Free+probability+theory%22">Free probability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+chaos%22">Quantum chaos</searchLink><br /><searchLink fieldCode="DE" term="%22Mean+field+theory%22">Mean field theory</searchLink><br /><searchLink fieldCode="DE" term="%22Rings+of+integers%22">Rings of integers</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n − θ( d) where θ( d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n −2( θ(2)+ θ( d)). For Weyl polynomials and Binomial polynomials, this probability decays respectively like $\exp{(-2\theta_{\infty}}\sqrt{n})$ and $\exp{(-\pi\theta _{\infty}\sqrt{n})}$ where θ ∞ is such that $\theta(d)=2^{-3/2}\theta_{\infty}\sqrt{d}$ in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [ a, b] has a scaling form given by $\exp{(-N_{ab}\tilde{\varphi}(k/N_{ab}))}$ where N ab is the mean number of real roots in [ a, b] and $\tilde{\varphi}(x)$ a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, ). [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/s10955-008-9574-3 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 39 StartPage: 235 Subjects: – SubjectFull: Random polynomials Type: general – SubjectFull: Heat equation Type: general – SubjectFull: Free probability theory Type: general – SubjectFull: Quantum chaos Type: general – SubjectFull: Mean field theory Type: general – SubjectFull: Rings of integers Type: general Titles: – TitleFull: Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Schehr, Grégory – PersonEntity: Name: NameFull: Majumdar, Satya N. IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 07 Text: Jul2008 Type: published Y: 2008 Identifiers: – Type: issn-print Value: 00224715 Numbering: – Type: volume Value: 132 – Type: issue Value: 2 Titles: – TitleFull: Journal of Statistical Physics Type: main |
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