HEXAGONbeta

Explicit onsets, rates and the tail of the collision constant for heat-evolved Weyl polynomials

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

Contributed by Bingo Baggins ↗

Version 1 / Oct 23, 2026 / CC0 1.0

Abstract

Let hkh_k be the orthonormal Hermite polynomials for the standard Gaussian weight, and let a(w)=∑j≥0ajwja(w)=\sum_{j\ge0}a_jw^j be a real power series with radius of convergence greater than 11, a(0)≠0a(0)\ne0, no zeros on the unit circle and only simple zeros in the unit disk. For all large nn the polynomial ∑j=0najhn−j\sum_{j=0}^{n}a_jh_{n-j} has exactly as many nonreal zeros as aa has in the disk, but the known proof gives the onset by compactness. We make it explicit: if aa has no zeros in {1−d≤∣w∣≤1}\{1-d\le|w|\le1\}, with dd below an explicit threshold, the count is exact as soon as nn exceeds a polynomial in 1/d1/d and in the inverses of ∣a(0)∣|a(0)|, min⁡∣w∣=1∣a∣\min_{|w|=1}|a|, min⁡∣w∣=1−d∣a∣\min_{|w|=1-d}|a| and min⁡[−1+d,1−d]max⁡(∣a∣,∣a′∣)\min_{[-1+d,1-d]}\max(|a|,|a'|); no separation of the nonreal zeros of aa is needed. By an exact identity this applies to the random Weyl polynomials GnG_n with independent standard Gaussian coefficients under the backward heat flow Gn,t=exp⁡(−t2n∂z2)GnG_{n,t}=\exp(-\frac t{2n}\partial_z^2)G_n, whose real-rooting time converges in distribution to T∗=R∗−2T_*=R_*^{-2}, where R∗R_* is the smallest modulus of a nonreal zero of the Gaussian power series q(w)=∑jηjwjq(w)=\sum_j\eta_jw^j. We show that the convergence holds with the rate n−αn^{-\alpha} on [1+ε,∞)[1+\varepsilon,\infty) for every α<350\alpha<\frac3{50}, an exponent that does not depend on ε\varepsilon; the dependence on ε\varepsilon is removed by a super-exponential tail for the number of zeros of qq in a disk. Under a coupling of the degrees that shares the top coefficients, the real-rooting time Λn\Lambda_n satisfies ∣Λn−T∗∣≤K/n|\Lambda_n-T_*|\le K/n for n≥n0n\ge n_0, with KK and n0n_0 depending on qq. We bound KK and n0n_0 by polynomials in condition parameters of qq, with absolute degrees and, when T∗≥1+εT_*\ge1+\varepsilon, coefficients polynomial in 1/ε1/\varepsilon. For the Gaussian power series this gives, on the event {T∗≥1+ε}\{T_*\ge1+\varepsilon\}, a power tail with exponent 1/9001/900 for the largest scaled displacement DD of the final collision of zeros, and without restriction the tail bound exp⁡(−c(log⁡x/log⁡log⁡x)1/2)\exp(-c(\log x/\log\log x)^{1/2}); the inputs include closed forms of the two-point correlation functions of the zeros of qq.

References

  1. B. Baggins, The real-rooting time of heat-evolved random polynomials , preprint (2026).
  2. B. Baggins, Nonreal zeros of heat-evolved Weyl polynomials near the collapse time , preprint (2026).
  3. B. Baggins, Exact counts of nonreal zeros beyond the Weyl weight: varying weights, coefficient profiles and the Laguerre flow , preprint (2026).
  4. T. W. Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities , Proc. Amer. Math. Soc. 6 (1955), 170--176.
  5. J.-M. Azaïs, M. Wschebor, Level Sets and Extrema of Random Processes and Fields , Wiley, Hoboken, NJ, 2009.
  6. P. J. Forrester, The limiting Kac random polynomial and truncated random orthogonal matrices , J. Stat. Mech. Theory Exp. (2010), P12018; doi:10.1088/1742-5468/2010/12/P12018; arXiv:1009.3066.arXivDOI
  7. M. Fujiwara, Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung , Tôhoku Math. J. 10 (1916), 167--171.
  8. M. Kac, On the average number of real roots of a random algebraic equation , Bull. Amer. Math. Soc. 49 (1943), 314--320.
  9. G. Lindgren, K. Podgórski, I. Rychlik, Effective computations of joint excursion times for stationary Gaussian processes , arXiv:2007.14220 (2020).arXiv
  10. S. Matsumoto, T. Shirai, Correlation functions for zeros of a Gaussian power series and Pfaffians , Electron. J. Probab. 18 (2013), no. 49, 1--18; doi:10.1214/EJP.v18-2545; arXiv:1212.6108.arXivDOI

Version history

  1. v1Submitted by Bingo BagginsInitial depositCurrentOct 22, 2026