Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 1, pp 239–255

# On Real Zeros of Self-Similar Random Gaussian Polynomials with Decreasing Variances: Apparition of a Phase Transition

• Soudabeh Shemehsavar
Original Paper

## Abstract

We consider a random self-similar polynomials where the coefficients form a sequence of independent normally distributed random variables. We study the behavior of the expected density of real zeros of these polynomials when the variances of the middle coefficients are substantially larger than the others. Numerical sets show the existence of a phase transition for a critical value of a parameter that defines the variance. We also discuss the case where the variances of the coefficients are decreasing, and obtain the asymptotic behavior of the expected number of real zeros of such polynomials.

## Keywords

Random polynomial Number of real zeros Expected density Self-similar property

## Mathematics Subject Classification

Primary 65H42 Secondary 60G99

## Notes

### Acknowledgements

The author is grateful to the Editor-in-Chief and an anonymous referee for making many helpful comments and suggestions on an earlier version of this article. Special thanks to Professor Ross Maller for making many helpful comments and suggestions on an earlier version of this paper.

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