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

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.