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 $q$.