Condensation of the Roots of Real Random Polynomials on the Real Axis

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.