Hexagon

math.PR — Probability

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

Contributed by Bingo Baggins

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

Primarily AI-generated textHuman understanding: no partsHermite polynomialsWeyl polynomialscollision of zeroscorrelation functions of zerosexplicit onsetheat flowlinear combinations of orthogonal polynomialsnonreal zerosrandom power seriesrate of convergence

Oct 23, 2026PDF ↗View work →

hexagon:2610.00483v1

math.PR — Probability

The Real-Rooting Time Of Heat-Evolved Random Polynomials

Contributed by Bingo Baggins

Let $G_{n,t}=\exp(-\frac{t}{2n}\partial_z^2)G_n$ be a random Weyl polynomial of degree $n$ 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 $t=1$ . We show that the first time at which all zeros of $G_{n,t}$ are real, a finite-degree analogue of the de Bruijn--Newman constant, converges in distribution to $T_*=R_*^{-2}$ , where $R_*$ is the smallest modulus of a nonreal zero of the random power series $\sum_j\eta_jw^j$ 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 $[1,\infty]$ . The proof rests on an exact count of the nonreal zeros of analytic combinations $\sum_{j\le n}a_jh_{n-j}$ of orthonormal Hermite polynomials: if the real power series $a$ has radius of convergence greater than $1$ , no zeros on the unit circle and simple zeros in the unit disk, then for all large $n$ the combination has exactly as many nonreal zeros as $a$ has in the disk. Its method expresses the polynomial and its derivative in the bulk through a $2\times2$ matrix applied to $(h_n,h_{n-1})$ , built from only $O(\log n)$ steps of the backward three-term recurrence, with the remaining terms controlled by a Christoffel--Darboux bound. For Gaussian coefficients the law of $T_*$ is a Fredholm determinant, $\mathbb{P}(T_*>t)\sim\frac4{3\pi}t^{-3/2}$ as $t\to\infty$ , and $(t-1)\log\mathbb{P}(T_*\le t)\to-\pi^2/12$ as $t\downarrow1$ , half of the constant for complex coefficients. At the scale $1/n$ the last pair of nonreal zeros collides inside the real bulk, at relative position $\cos\theta_*$ , where $R_*e^{\pm i\theta_*}$ are the nonreal zeros of minimal modulus, and the real-rooting time is within $O(1/n)$ of $T_*$ . In a window of size $1/n$ the polynomial is, up to a positive factor and $O(1/n)$ , the model $\zeta\cos(\zeta+c)+\lambda\sin(\zeta+c)$ , whose threshold for real-rootedness is given by Kepler's equation; the phase $c$ rotates with $n$ , so $n(\Lambda_n-T_*)$ oscillates over an explicit interval and has an explicit limit law jointly with $T_*$ .

Primarily AI-generated textHuman understanding: no partsFredholm determinantsHermite polynomialsPfaffian point processesWeyl polynomialscollision of zerosde Bruijn-Newman constantheat flowlinear combinations of orthogonal polynomialsrandom power seriesreal zeros

Oct 22, 2026PDF ↗View work →

Submitted works

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

The Real-Rooting Time Of Heat-Evolved Random Polynomials