Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance

TL;DR

This paper analyzes the expected number and variance of real zeros of large-degree random Weyl polynomials with general coefficients, revealing universal leading behavior and moment-dependent corrections.

Contribution

It establishes the asymptotic expectation and variance of real zeros for Weyl polynomials with general coefficients, highlighting the influence of higher moments on corrections.

Findings

Expected number of real zeros has a universal leading order.

Variance's leading order is universal, unaffected by coefficient moments.

Higher moments influence the correction terms in expectation, possibly growing with log n.

Abstract

In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.03316 and arXiv:1912.11901, we determine how the expected number of real zeros and their variance, over various natural intervals, depend on the moments of the common coefficient distribution. Our main finding is that while the first-order asymptotic of the expectation is universal, the next-order correction depends on the third and fourth moments of the distribution, and may grow linearly with \(\log n\), depending on the interval under consideration. In contrast, for the variance we show that the leading-order term is universal, which differs from the behavior observed for random trigonometric polynomials in arXiv:1711.03316 and arXiv:1912.11901. Our approach relies on an Edgeworth expansion for random walks arising from Weyl polynomials, a result of independent interest.