2. Other properties of the zero measures
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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.) -
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 $E$.
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 $N_n(E)$. -
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
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. -
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.