The Annals of Probability

No zero-crossings for random polynomials and the heat equation

Amir Dembo and Sumit Mukherjee

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Abstract

Consider random polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ 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 $\alpha=d/2-1$.

Article information

Source
Ann. Probab., Volume 43, Number 1 (2015), 85-118.

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1415801553

Digital Object Identifier
doi:10.1214/13-AOP852

Mathematical Reviews number (MathSciNet)
MR3298469

Zentralblatt MATH identifier
1312.60036

Subjects
Primary: 60G15: Gaussian processes 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
Secondary: 35K05: Heat equation 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48]

Keywords
Random polynomials real zeros heat equation Gaussian processes regularly varying

Citation

Dembo, Amir; Mukherjee, Sumit. No zero-crossings for random polynomials and the heat equation. Ann. Probab. 43 (2015), no. 1, 85--118. doi:10.1214/13-AOP852. https://projecteuclid.org/euclid.aop/1415801553


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