On expected number of level crossings of a random hyperbolic polynomial

Abstract

Let $g_1(\omega),g_2(\omega),\ldots, g_n(\omega)$ be independent and normally distributed random variables with mean zero and variance one. We show that, for large values of $n$, the expected number of times the random hyperbolic polynomial $y=g_1(\omega)\cosh x+ g_2(\omega)\cosh 2x+\cdots +g_n(\omega)\cosh nx$ crosses the line $y=L$, where $L$ is a real number, is $\frac{1}{\pi}\log n +O(1)$ if $L=o(\sqrt{n})$ or ${L}/{\sqrt{n}} =O(1)$, but decreases steadily as $O(L)$ increases in magnitude and ultimately becomes negligible when $n^{-1}\log {L}/{\sqrt{n}}\to\infty$.