Universality for zeros of random polynomials

Thomas Bloom, Duncan Dauvergne

Published 2018-01-30Version 1

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau$ satisfying the Bernstein-Markov property on a regular compact set $K \subset \mathbb{C}$. We show that if $\mathbb{P}(|\xi_0|>e^{|z|})=o(\frac{1}{|z|})$, then the normalized counting measure of the zeros of $H_n$ converges in probability to the equilibrium measure of $K.$ This is the best possible result, in the sense that the roots of $H_n(z)=\sum_{j=0}^n\xi_jz^j$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $\xi_j$ is not satisfied. In addition, we give a multivariable version of this result. We also consider random polynomials of the form $\sum_{j=0}^n\xi_kf_{n,k}z^k$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{\xi_k\}$ satisfy $\mathbb{E} \log(1 + |\xi_0|) < \infty$. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.