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3. Extreme values

    1. 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)?
          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.
        • 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.