2020.01.31 PS - Marcus Michelen  - Zero free regions and central limit theorems
Channel: Courant Institute Seminars
Date: 2020-02-18

Let X be a random variable taking values in {0,...,n} and f(z) be its probability generating function.  Pemantle conjectured that if the variance of X is large and f has no roots close to 1 in the complex plane, then X must be approximately normal. We will discuss a complete resolution of this conjecture in a strong quantitative form, thereby giving the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. Additionally, if f has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form. These results also imply a multivariate central limit theorem that answers a conjecture and completes a program of Ghosh, Liggett and Pemantle.  This talk is based on joint work with Julian Sahasrabudhe.