No zero-crossings for random polynomials and the heat equation

TL;DR

This paper investigates the zero-crossings of random polynomials with Gaussian coefficients and the heat equation solutions, establishing probability decay rates for the absence of roots over specific intervals and connecting these to persistence probabilities.

Contribution

It provides new criteria for the continuity of persistence exponents and applies these to determine the probability of no roots in certain intervals for random polynomials and heat equation solutions.

Findings

Probability of no roots in [0,1] scales as n^{-b_{α}+o(1)}

Probability of no roots in (1,∞) scales as n^{-b_0+o(1)}

Heat equation solution remains non-zero over time with probability T^{-b_{α}+o(1)}

Abstract

Consider random polynomial $\sum_{i=0}^na_ix^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $\alpha$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_{\alpha}+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_{\alpha}-2b_0+o(1)}$. Here, $b_{\alpha}=0$ when $\alpha\le-1$ and otherwise $b_{\alpha}\in(0,\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $\phi_d({\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $\phi_d({\mathbf{x}},0)$, we confirm that the probability of $\phi_d({\mathbf{x}},t)\neq0$ for all $t\in[1,T]$, is $T^{-b_{\alpha}+o(1)}$, for $\alpha=d/2-1$.