Real roots near the unit circle of random polynomials

Let f n ( z stretchy="false">) = ∑ k 0 ε f_n(z) = \sum _{k 0}^n \varepsilon _k z^k be a random polynomial where alttext="epsilon comma ellipsis Baseline"> , …<!-- … encoding="application/x-tex">\varepsilon _0,\ldots ,\varepsilon _n are i.i.d. variables with alttext="double-struck upper E 1 0"> E 1 encoding="application/x-tex">\mathbb {E} _1 0 and squared 1"> 2 _1^2 1 . Letting alttext="r r 2 r encoding="application/x-tex">r_1, r_2,\ldots , r_k denote the real roots of n"> encoding="application/x-tex">f_n , we show that point process defined by alttext="StartSet StartAbsoluteValue EndAbsoluteValue minus EndSet"> fence="false" stretchy="false">{ stretchy="false">| −<!-- − stretchy="false">} encoding="application/x-tex">\{|r_1| - 1,\ldots |r_k| \} converges to non-Poissonian limit on scale alttext="n negative encoding="application/x-tex">n^{-1} as right-arrow normal infinity"> stretchy="false">→<!-- → mathvariant="normal">∞<!-- ∞ encoding="application/x-tex">n \to \infty Further, for each alttext="delta greater-than δ<!-- δ <mml:mo>&gt; encoding="application/x-tex">\delta &gt; has root within alttext="normal Theta delta slash right-parenthesis"> mathvariant="normal">Θ<!-- Θ <mml:mo>/ encoding="application/x-tex">\Theta _{\delta }(1/n) unit circle probability at least alttext="1 delta"> encoding="application/x-tex">1 \delta This resolves conjecture Shepp Vanderbei from 1995 confirming its weakest form refuting strongest form.