Level crossings and turning points of random hyperbolic polynomials (Q1587818)

For real \(x\), let \(T_1(x):=\sum_{j=1}^n\eta_j\cosh jx\) and \(T_2(x):=\sum_{j=1}^n\eta_j\sinh jx\), where \(\eta_1,\dots,\eta_n\) are independent standard normal random variables. For \(i=1,2\) and a constant \(K\), let \(N_K^{(i)}\) be the number of real roots of the equation \(T_i(x)=K\), and let \(M^{(i)}\) be the number of real zeros of \(T_i'(x)\) (the first derivative of \(T_i(x)\)). It is proved that the expectation \(EM^{(1)}\sim\pi^{-1}\log n\) as \(n\to\infty\), and the expectation \(EN_K^{(2)}\sim\pi^{-1}\log n\) as \(n\to\infty\) provided a constant \(K=o(\sqrt{n\log n})\) as \(n\to\infty\). Similar results for \(EM^{(2)}\) and \(EN_K^{(1)}\) have been proved earlier.