Real roots of Random Polynomials: Universality close to accumulation points

TL;DR

This paper investigates the local behavior of real roots near accumulation points in large-degree random polynomials, revealing universal density forms and the effects of coefficient mean on root suppression.

Contribution

It identifies a universal scaling region near accumulation points and demonstrates how the mean of coefficients influences root density suppression.

Findings

Density of roots tends to a universal form near accumulation points.

Suppression of real roots occurs gradually with nonzero mean coefficients.

Universal behavior holds for polynomials with finite second moment of coefficients.

Abstract

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent, identically distributed coefficients c_i, as long as the second moment \sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to the previously reported abrupt) and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled as \mu_n\sim n^{-1/2}.