Sums of random polynomials with independent roots

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:495

Article

O'Rourke, Sean¹  Reddy, Tulasi Ram² 

[1] Univ Colorado, Dept Math, Campus Box 395, Boulder, CO 80309 USA

[2] New York Univ Abu Dhabi, Abu Dhabi, U Arab Emirates

关键词: Random polynomials;    Logarithmic potential;    Zeros of sums of polynomials;    Anti-concentration;   

DOI  :  10.1016/j.jmaa.2020.124719

来源: Elsevier

【 摘 要 】

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let p and q be two independent random polynomials of degree n, whose roots are chosen independently from the probability measures mu and nu in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum p + q as n tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of mu and nu. More generally, we consider the sum of m independent degree n random polynomials when m is fixed and n tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free