3. Extreme values
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Minimum modulus for Kac polynomials
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.