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# Maximumof Polynomials on the Unit Circle

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I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with uniform distribution on the unit circle. 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 $n\to\infty$ $$m\geq \exp(\alpha\sqrt{n}).$$ Or at least that for every $\epsilon>0$ there exists $n$ sufficiently large such that $$\mathbb{P}(m\geq\exp(\alpha\sqrt{n}))>1-\epsilon$$ for some $\alpha$ independent on $n$.

Any idea of what can be useful here?

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# Maximum of Polynomials on the Unit Circle

I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with uniform distribution on the unit circle. 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 $n\to\infty$ $$m\geq \exp(\alpha\sqrt{n}).$$

Any idea of what can be useful here?