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Limit law for root separation in random polynomials

Marcus Michelen, Oren Yakir

TL;DR

This work establishes a universal limit law for the minimal separation of roots of random polynomials with iid, mean-zero, sub-Gaussian coefficients: after scaling by $n^{5/4}$, all pairwise root distances converge in distribution to a non-homogeneous Poisson point process with intensity proportional to $t^3$. The authors develop a comprehensive net-based approach near the unit circle, combine refined small-ball bounds and Gaussian comparisons, and show that interior disk contributions vanish with a separate almost-sure result for random Taylor series. The combination yields a precise $n^{-5/4}$ scaling and an explicit limiting intensity constant, with significant implications for root repulsion and universal root statistics in random polynomials. The methods blend local analytic control, probabilistic anti-concentration, and Poisson-approximation techniques, offering a robust framework for analyze near-collision events in complex-root ensembles.

Abstract

Let $f_n$ be a random polynomial of degree $n\ge 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of $f_n$ and prove that the set of these distances, normalized by $n^{-5/4}$, converges in distribution as $n\to \infty$ to a non-homogeneous Poisson point process. As a corollary, we deduce that the minimal separation distance between roots of $f_n$, normalized by $n^{-5/4}$ has a non-trivial limit law. In the course of the proof, we establish a related result which may be of independent interest: a Taylor series with random i.i.d. coefficients almost-surely does not have a double zero anywhere other than the origin.

Limit law for root separation in random polynomials

TL;DR

This work establishes a universal limit law for the minimal separation of roots of random polynomials with iid, mean-zero, sub-Gaussian coefficients: after scaling by , all pairwise root distances converge in distribution to a non-homogeneous Poisson point process with intensity proportional to . The authors develop a comprehensive net-based approach near the unit circle, combine refined small-ball bounds and Gaussian comparisons, and show that interior disk contributions vanish with a separate almost-sure result for random Taylor series. The combination yields a precise scaling and an explicit limiting intensity constant, with significant implications for root repulsion and universal root statistics in random polynomials. The methods blend local analytic control, probabilistic anti-concentration, and Poisson-approximation techniques, offering a robust framework for analyze near-collision events in complex-root ensembles.

Abstract

Let be a random polynomial of degree whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of and prove that the set of these distances, normalized by , converges in distribution as to a non-homogeneous Poisson point process. As a corollary, we deduce that the minimal separation distance between roots of , normalized by has a non-trivial limit law. In the course of the proof, we establish a related result which may be of independent interest: a Taylor series with random i.i.d. coefficients almost-surely does not have a double zero anywhere other than the origin.
Paper Structure (45 sections, 33 theorems, 518 equations, 1 figure)

This paper contains 45 sections, 33 theorems, 518 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Heuristic explanation for the scaling exponent
  3. Double roots of random polynomials
  4. Double zeros of random analytic functions in the disk
  5. Outline of the proof
  6. Organization of the paper and notation
  7. Almost sure result in the disk: Proof of Theorem \ref{['thm:almost_sure_double_zeros_in_disk']}
  8. Preparations
  9. The perturbative argument
  10. Reducing Theorem \ref{['thm:poisson_limit_for_close_roots']} to a net argument
  11. Making use of the sub-Gaussian assumption
  12. "Worst case" small-ball probability bounds
  13. A small-ball probability bound for smooth points
  14. Gaussian comparison near the unit circle
  15. Reducing to an annulus: Proof of Proposition \ref{['prop:no_close_double_roots_in_the_bulk']}
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $f_n$ be given by eq:intro_def_f_n, and suppose $\xi_0$ is a mean-zero, sub-Gaussian random variable satisfying $\mathbb P[\xi_0 = 0] = 0$. Then the point process converges in distribution (with respect to the vague topology) as $n\to \infty$ to a non-homogeneous Poisson point process on $\mathbb R_{\ge 0}$ with intensity $\mathfrak{c}_\ast t^3$, for some $\mathfrak{c}_\ast >0$.

Figures (1)

  • Figure 1: Left: roots on the unit circle of $\text{Re}(f_n)$ for random Littlewood $f_n$ (red points); Right: the same number of i.i.d. uniform points on the unit circle (blue points).

Theorems & Definitions (112)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Claim 2.1
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • ...and 102 more