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Large N Program Series: Boris Hanin, MIT

Title: Pairing between zeros and critical points of random polynomials
Program Website: Large N limit problems in Kahler Geometry
Speaker: Boris Hanin, MIT
Date: Tuesday, June 02, 2015
Time: 10:00am – 11:00am
Place: Seminar Room 313, Simons Center
 

 

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Abstract: Consider a polynomial p_N(z) in one complex variable. The Gauss-Lucas Theorem says that the critical points of p_N lie inside the convex hull of its zeros. But are critical points actually distributed inside the convex hull if p_N is chosen at random?

The purpose of this talk is to explain that in fact each critical point of p_N typically comes paired with a single zero. The distance between a critical point and its paired zero is on the order of N^{-1}, which is much smaller than the typical N^{-1/2} spacing between order of N iid points on the sphere.

In the first part of my talk, I will give a heuristic interpretation for this pairing by relating zeros and critical points to electrostatics on the Riemann sphere. In the second part, I will explain how to use Bergman kernel asymptotics to prove rigorous theorems about this pairing.