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] |
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| Database: | Engineering Source |
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