Condensation of the roots of real random polynomials on the real axis

Details Condensation of the roots of real random polynomials on the real axis Condensation of the roots of real random polynomials on the real axis

2009-02-06

arxiv.org/abs/0902.1027v1

J. Stat. Phys. 135, 587-598 (2009)

Physics

Mathematical Physics

13 pages, 2 figures

Scientific paper

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance = e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly the mean number of real roots for large n. As \alpha is varied, one finds three different phases. First, for 0 \leq \alpha < 1, one finds that \sim (\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase where < N_n > grows algebraically with a continuously varying exponent, < N_n > \sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for \alpha > 2, one finds a third phase where \sim 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 are real. This condensation occurs via a localization of the real roots around the values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.

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