Abstract
We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree at most n associated with varying weights , with Gaussian coefficients. We deduce asymptotics of the variance for fixed exponential weights . In particular, we show that very generally, the variance is asymptotic to , where the constant C involves a universal constant and an equilibrium density associated with the weight(s).
Funding Statement
D. Lubinsky was partially supported by NSF grant DMS1800251. I. Pritsker was partially supported by NSA grant H98230-21-1-0008, and by the Vaughn Foundation endowed Professorship in Number Theory.
Acknowledgments
The authors would like to thank Yen Do, Hoi Nguyen, and Oanh Nguyen for their perspectives on random polynomials and comments on this paper. This paper grew out of the AIM workshop on Zeros of Random Polynomials, held from 12-16 August 2019.
Citation
Doron S. Lubinsky. Igor E. Pritsker. "Variance of real zeros of random orthogonal polynomials for varying and exponential weights." Electron. J. Probab. 27 1 - 32, 2022. https://doi.org/10.1214/22-EJP802
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