Maximum of Polynomials in the Unit Circle Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by
$$
P(z)=\prod_{i=1}^{n}(z-z_i).
$$
Let $m$ be the maximum absolute value of $P(z)$ on the unit circle $m=\max\{|P(z)|:|z|=1\}$. 
How can I estimate $m$? More specifically, I would like to prove that there exist $\alpha>0$ such that the following holds almost surely as $n\to\infty$
$$
m\geq e^{\alpha\sqrt{n}}.
$$ 
Any idea of what can be useful here?
 A: The logarithm of your objective function is
$$\sum_{i=1}^n \ln\lvert z - z_i\rvert\;.$$
If you think of the points as continuously distributed for large $n$, this becomes $n$ times the average
$$\frac{1}{\pi}\int_{D_1} \ln\lvert x-x' \rvert \mathrm d^2x'\;.$$
This is the potential of a uniformly charged disk at distance $|x|$ from the centre, within the disk. It can be evaluated, as in the three-dimensional case of a uniformly charged sphere, by splitting the integral into a part for the rings interior to $x$, for which the potential has to be evaluated at radius $|x|$, and a part for the rings exterior to $x$, for which the potential has to be evaluated at the radius of the rings:
$$
\begin{eqnarray}
\frac{1}{\pi}\int_{D_1} \ln\lvert x-x' \rvert \mathrm d^2x'
&=&
2\left(\int_0^{|x|}\ln|x|r\mathrm dr+\int_{|x|}^1\ln r r\mathrm dr\right)\\
&=&\frac{1}{2}\left(|x|^2-1\right)\;.
\end{eqnarray}
$$
So the average contribution from a uniform density of points is zero at the boundary and negative everywhere else. Does that fit with your numerical experiments? It suggests that the maximum should typically be near the boundary for large $n$.
It also suggests that you can think of the square root term in your conjecture as arising from fluctuations around this uniform density, so the conjecture could be rephrased as something like "Almost surely, there will be local fluctuations in the density of the points to create a fluctuation of $\alpha\sqrt{n}$ in the sum of the logarithms at some point on the boundary." I'll give some more thought to how this might be made rigorous.
A: I believe the conjecture that $\sup_\theta \prod_j |\theta -  z_j| \gg e^{c\sqrt{n}}$ almost surely is not true. It would imply $\sup_\theta \sum_j \log |\theta - z_j| \gg \sqrt{n}$. But from the analogy with the absolute value of a Brownian motion $|B_t|$, $t \in [0,1]$, I think one can only say $\sup_\theta \frac{1}{\sqrt{n}}\sum_j \log |\theta - z_j| > 0$ almost surely.
Consider the stochastic process $F(\zeta) = \frac{1}{\sqrt{n}}\sum_j \log|\zeta - z_j|$, where $|z_j| = 1$ and $|\zeta| = 1$. I claim $F$ converges a Gaussian process, in the sense that any finite dimensional distribution is jointly Gaussian. This can be established with moment method of CLT, using only mixed 2nd moments, since those terms dominate the sum for small moment powers, together with the fact that $\frac{1}{2\pi} \int_0^{2\pi} (\log(2(1- \cos t)))^k dt \ll k! / 2^k$, so that the tail of the Taylor expansion vanishes. The covariance structure of this process doesn't seem to admit closed form, but my Mathematica plot shows it's convex with value between 1 and $-1/2$ (for two antipodal $\zeta$'s).
$$ G(\eta) := \frac{1}{2\pi} \int_0^{2\pi} \log(2(1 + \cos \theta)) \log(2(1 + \cos(\theta + \eta))) d \theta.$$
$$ C(\eta) := G(\eta) / G(0); \qquad G(0) = \frac{\pi^2}{3}.$$
Now we can borrow ideas from James Lee's lecture notes to get an upper bound of $\sup F(\zeta)$ by Slepian comparison with some better understood Gaussian processes.

