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

TL;DR

This paper studies a family of real random polynomials with Gaussian coefficients, revealing phase transitions in the number of real roots and showing a condensation phenomenon where many roots localize on the real axis as degree increases.

Contribution

It provides an exact computation of the mean number of real roots for large degree and identifies three distinct phases with different growth behaviors based on the parameter lpha.

Findings

For 0 lpha<1, ext{mean roots} \u2264 (rac{2}{\u03c0}) ext{logarithmic growth}

For 1 lpha<2, ext{mean roots} ext{ grow algebraically with } n^{\u00alpha/2}

For lpha>2, ext{mean roots} ext{ grow linearly with } n

Abstract

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance <a_k^2> = e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly the mean number of real roots <N_n> for large n. As \alpha is varied, one finds three different phases. First, for 0 \leq \alpha < 1, one finds that <N_n> \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 <N_n> \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_n>/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.