Correlations between zeros of a random polynomial

TL;DR

This paper derives exact formulas for correlations between zeros of Kac and SO(2) random polynomials, revealing asymptotic independence and translation-invariant properties of zeros within specific intervals.

Contribution

It provides the first exact analytical expressions for zero correlations in Kac and SO(2) polynomials, highlighting their asymptotic behaviors and invariance properties.

Findings

Zeros in (-1,1) become asymptotically independent of zeros outside this interval.

Straightened zeros exhibit translation-invariant correlations in the limit.

Exact correlation formulas are obtained for both Kac and SO(2) polynomials.

Abstract

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval $(-1,1)$ are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random polynomial.