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 Sib. Èlektron. Mat. Izv., 2019, Volume 16, Pages 217–228 (Mi semr1052)

Real, complex and functional analysis

The Kostlan–Shub–Smale random polynomials in the case of growing number of variables

V. Gichev

Sobolev Institute of Mathematics, Omsk Branch 13, Pevtsova str., Omsk, 644099, Russia

Abstract: Let $\mathcal{P}_n=\sum_{j}\mathcal{H}_{j}$ be the decomposition in $L^2(S^m)$ of the space of homogeneous polynomials of degree $n$ on $\mathbb{R}^{m+1}$ into the sum of irreducible components of the group $\mathrm{SO}(m+1)$. We consider the asymptotic behavior of the sequence $\nu_{n}(t)=\frac{\mathsf{E}(|\pi_{j}u|^{2})}{\mathsf{E}(|u|^{2})}$, where $t=\frac{j}{n}$, $\pi_{j}$ is the projection onto $\mathcal{H}_{j}$, and $\mathsf{E}$ stands for the expectation in the Kostlan-Shub–Smale model for random polynomials. Assuming $\frac{m}{n}\to a>0$ as $n\to\infty$, we prove that $\nu_{n}(t)$ is asymptotic to $\sqrt{\frac{4+a}{\pi n}} e^{-n(1+\frac{a}{4})(t-\sigma_{a})^{2}}$, where $\sigma_{a}=\frac12(\sqrt{a^{2}+4a}-a)$.

Keywords: random polynomials.

 Funding Agency Grant Number Siberian Branch of Russian Academy of Sciences 1.1.1.4, project No. 03-14-2016-0004 The work is supported by the program of fundamental researches of SBRAS No. 1.1.1.4, project No. 03-14-2016-0004.

DOI: https://doi.org/10.33048/semi.2019.16.013

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UDC: 517.58
MSC: 43A85
Received June 23, 2017, published February 8, 2019
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Citation: V. Gichev, “The Kostlan–Shub–Smale random polynomials in the case of growing number of variables”, Sib. Èlektron. Mat. Izv., 16 (2019), 217–228

Citation in format AMSBIB
\Bibitem{Gic19} \by V.~Gichev \paper The Kostlan--Shub--Smale random polynomials in the case of growing number of variables \jour Sib. \Elektron. Mat. Izv. \yr 2019 \vol 16 \pages 217--228 \mathnet{http://mi.mathnet.ru/semr1052} \crossref{https://doi.org/10.33048/semi.2019.16.013} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000462268100013} `