Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

2. Other properties of the zero measures


    1. Add remark

      Problem 2.1.

      [Raphael Butez] Let P_n(z) = \sum_{0\le k\le n} a_k R_k^{(n)}(z), where the a_k are iid random variables satisfying \mathbb E \log(1+|a_0|)<\infty, and R_k^{(n)} are orthonormal under the scalar product \langle p,q\rangle = \int p(z) \overline{q(z)} e^{-2nV(z)} d\nu(z) for some “regular" measure \nu.

      Prove an LDP for the empirical measure of zeros under the assumption that the a_k have a density g with sub-Gaussian tails and vanishing like |z|^\alpha at z=0.

      (Note the LDP has recently been established by Butez and Zeitouni under the assumption that g is uniformly positive in a neighborhood of zero.)


        • Add remark

          Problem 2.2.

          [Thomas Bloom & Ofer Zeitouni] For a sequence of random orthogonal polynomials P_n, fix a set E\subset \mathbb C compactly contained in the complement of the limiting support of the empirical measure of zeros of P_n, and let N_n(E) be the number of zeros contained in E. Establish upper tail ("overcrowding") estimates on N_n(E).
              Note this should be easy for the special case of Gaussian Kac polynomials, where the limiting support is the unit circle. Indeed, for any compact subset of the unit disk, the zero process of P_n can be approximated by that of the limiting Gaussian analytic function F(z), which is a determinantal point process with Bergman kernel. In particular, the number of zeros of F inside E can be represented as a sum \sum_{k\ge 1} B_k of independent Bernoulli variables with expectation \lambda_k, where \{\lambda_k\} is the spectrum for the restriction of the Bergman kernel to E.


            • Add remark

              Problem 2.3.

              [Ofer Zeitouni] Let P_n(z) be a random Kac polynomial with iid coefficients that are not necessarily Gaussian. Consider the rescaled point process of zeros in a region within O(1/n) of the unit circle (blown up by n to live at scale order 1). Show the correlation functions for this point process are asymptotically universal as n\to \infty.

                • Universality for local statistics of zeros in the bulk


                  Add remark

                  Problem 2.4.

                  [Raphael Butez] Consider an array of polynomials \{p_{n,k}\}_{1\le k\le n<\infty } where for each n, \{p_{n,k}\}_{1\le k\le n} is an orthonormal set with respect to the inner product \langle p,q\rangle_n = \int p \overline q e^{-2nV_\nu}d\nu, where V_\nu is the logarithmic potential for the reference measure \nu, and the support S of \nu has nonempty interior. Put P_n(z) = \sum_{0\le k\le n} \xi_k p_{n,k}, where the \xi_k are iid (say Gaussian). Show that for any fixed z_0\in S^o, the point process of zeros of P_n, centered at z_0 and blown up by \sqrt{n}, converges in distribution to the zero set of the planar Gaussian analytic function.


                    • Add remark

                      Problem 2.5.

                      [Thomas Bloom] Let P_n(z) = \sum_{0\le k\le n} \xi_k p_{n,k}(z), where the \xi_k are iid non-degenerate random variables (with some additional conditions), and \{p_{n,k}\}_{1\le k\le n} is an orthonormal set with respect to the inner product \langle p,q\rangle_n = \int p \overline q e^{-2nV}d\nu. Show the empirical measures of zeros \mu_n for P_n converge weakly (in probability or almost surely) to some measure \mu.

                          Cite this as: AimPL: Zeros of random polynomials, available at http://aimpl.org/randpolyzero.