HEXAGONbeta

The Real-Rooting Time Of Heat-Evolved Random Polynomials

Primarily AI-generated textHuman understanding: no partsmath.PR — Probabilitymath.CA — Classical Analysis and ODEsmath.CV — Complex Variablesmath.FA — Functional Analysis

Contributed by Bingo Baggins ↗

Version 1 / Oct 22, 2026 / CC0 1.0

Abstract

Let Gn,t=exp⁡(−t2n∂z2)GnG_{n,t}=\exp(-\frac{t}{2n}\partial_z^2)G_n be a random Weyl polynomial of degree nn under the backward heat flow, with independent coefficients of a law that is not a point mass and has finite variance. Hall, Ho, Jalowy and Kabluchko showed that the empirical zero distribution collapses onto the real line at t=1t=1. We show that the first time at which all zeros of Gn,tG_{n,t} are real, a finite-degree analogue of the de Bruijn--Newman constant, converges in distribution to T∗=R∗−2T_*=R_*^{-2}, where R∗R_* is the smallest modulus of a nonreal zero of the random power series ∑jηjwj\sum_j\eta_jw^j with coefficients of the same law; the convergence is almost sure under a coupling of the degrees that shares the top coefficients, whereas under the standard coupling the set of limit points of the real-rooting time is almost surely [1,∞][1,\infty]. The proof rests on an exact count of the nonreal zeros of analytic combinations ∑j≤najhn−j\sum_{j\le n}a_jh_{n-j} of orthonormal Hermite polynomials: if the real power series aa has radius of convergence greater than 11, no zeros on the unit circle and simple zeros in the unit disk, then for all large nn the combination has exactly as many nonreal zeros as aa has in the disk. Its method expresses the polynomial and its derivative in the bulk through a 2×22\times2 matrix applied to (hn,hn−1)(h_n,h_{n-1}), built from only O(log⁡n)O(\log n) steps of the backward three-term recurrence, with the remaining terms controlled by a Christoffel--Darboux bound. For Gaussian coefficients the law of T∗T_* is a Fredholm determinant, P(T∗>t)∼43πt−3/2\mathbb{P}(T_*>t)\sim\frac4{3\pi}t^{-3/2} as t→∞t\to\infty, and (t−1)log⁡P(T∗≤t)→−π2/12(t-1)\log\mathbb{P}(T_*\le t)\to-\pi^2/12 as t↓1t\downarrow1, half of the constant for complex coefficients. At the scale 1/n1/n the last pair of nonreal zeros collides inside the real bulk, at relative position cos⁡θ∗\cos\theta_*, where R∗e±iθ∗R_*e^{\pm i\theta_*} are the nonreal zeros of minimal modulus, and the real-rooting time is within O(1/n)O(1/n) of T∗T_*. In a window of size 1/n1/n the polynomial is, up to a positive factor and O(1/n)O(1/n), the model ζcos⁡(ζ+c)+λsin⁡(ζ+c)\zeta\cos(\zeta+c)+\lambda\sin(\zeta+c), whose threshold for real-rootedness is given by Kepler's equation; the phase cc rotates with nn, so n(Λn−T∗)n(\Lambda_n-T_*) oscillates over an explicit interval and has an explicit limit law jointly with T∗T_*.

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Version history

  1. v1Submitted by Bingo BagginsInitial depositCurrentOct 22, 2026