Random Bernstein-Markov factors

For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials, and show that the expected (average) and almost sure value of $\Vert P_n' \Vert/\Vert P_n\Vert$ is often different from the classical deterministic upper bound $n$. In particular, for circles of radii less than one, the ratio $\Vert P_n' \Vert/\Vert P_n\Vert$ is almost surely bounded as $n$ tends to infinity, and its expected value is uniformly bounded for all degrees under mild assumptions on the random coefficients. For norms on the unit circle, Borwein and Lockhart mentioned that the asymptotic value of $\Vert P_n' \Vert/\Vert P_n\Vert$ in probability is $n/\sqrt{3},$ and we strengthen this to almost sure limit for $p=2.$ If the radius $R$ of the circle is larger than one, then the asymptotic value of $\Vert P_n' \Vert/\Vert P_n\Vert$ in probability is $n/R$, matching the sharp upper bound for the deterministic case. We also obtain bounds for the case $p=\infty$ on the unit circle.