Real roots near the unit circle of random polynomials

Abstract

Let f n ( z ) = ∑ k = 0 n ε k z k f_n(z) = \sum _{k = 0}^n \varepsilon _k z^k be a random polynomial where ε 0 , … , ε n \varepsilon _0,\ldots ,\varepsilon _n are i.i.d. random variables with E ε 1 = 0 \mathbb {E} \varepsilon _1 = 0 and E ε 1 2 = 1 \mathbb {E} \varepsilon _1^2 = 1 . Letting r 1 , r 2 , … , r k r_1, r_2,\ldots , r_k denote the real roots of f n f_n , we show that the point process defined by { | r 1 | − 1 , … , | r k | − 1 } \{|r_1| - 1,\ldots , |r_k| - 1 \} converges to a non-Poissonian limit on the scale of n − 1 n^{-1} as n → ∞ n \to \infty . Further, we show that for each δ > 0 \delta > 0 , f n f_n has a real root within Θ δ ( 1 / n ) \Theta _{\delta }(1/n) of the unit circle with probability at least 1 − δ 1 - \delta . This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form.