Anti-concentration and the geometry of polynomials

Abstract:   Let X be a random variable taking values in {0,...,n} with standard deviation sigma and let f_X be its probability generating function. Pemantle conjectured that if sigma is large and f_X has no roots close to 1 in the complex plane then X must approximate a normal distribution. In this talk, I will discuss the resolution of Pemantle's conjecture and its application to prove a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for so-called strong Rayleigh distributions. I will also touch on some more recent work connecting anti-concentration for random variables with the zeros of their probability generating functions.

This is based on joint work with Marcus Michelen.