3. Extreme values
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Minimum modulus for Kac polynomials
An upper bound $\mathbb P( \sqrt{n} m_n \le \epsilon) = O(\epsilon)$ was established by Konyagin and Schlag in the case that the $\xi_k$ are Rademacher.Problem 3.1.
[Hoi Nguyen] Consider random Kac polynomials $P_n(z) = \sum_{0\le k\le n} \xi_k z^k$ with $\xi_k$ iid standard Gaussians (real or complex), and put $m_n:= \min_{|z|=1} |P_n(z)|$. Does $\sqrt{n} m_n$ converge in distribution? Is the limit universal (same for non-Gaussian coefficients with the same normalization)? -
Non-Gaussian multiplicative chaoses
Problem 3.2.
[Ofer Zeitouni] Let $P_n(z) = \sum_{0\le k\le n} \frac1{\sqrt{k}} \xi_k z^k$ with $\xi_k$ iid centered random variables of variance one, and consider the random measures $\mu_n$ on $S^1$ with density $$ \exp( \gamma P_n(z) - \frac{\gamma^2}2 Var(P_n(z))) $$ with respect to Lebesgue measure, where $\gamma>0$ is a parameter.
When the $\xi_k$ are Gaussian it is known that for $\gamma$ sufficiently small these random measures converge weakly to a limiting random measure $\mu_\infty^\gamma$ (with respect to the vague topology) known as the Gaussian multiplicative chaos.
Prove convergence to a limiting random measure $\mu_\infty^\gamma$ in the general non-Gaussian case, and show that the ($\gamma$-dependent) Hausdorff dimension is the same as in the Gaussian case. -
Maximum modulus for log-correlated polynomials on the circle
Related to the previous problem on properties of the random measures $\mu_n$ is to determine the level of the maximum of the random "Hamiltonian" $P_n(z)$ (note that for large values of $\gamma$, near-maximizers will receive a lot of weight).Problem 3.3.
[Nick Cook] Let $P_n(z) = \sum_{0\le k\le n} \frac1{\sqrt{k}} \xi_k z^k$ with $\xi_k$ iid centered random variables of variance one and sub-exponential tails, and put $M_n = \max_{|z|=1} |P_n(z)|$. Prove a law of large numbers for $M_n$.
Cite this as: AimPL: Zeros of random polynomials, available at http://aimpl.org/randpolyzero.