Zeros of polynomials: Limiting distribution, exponential profiles and free probability
A talk in the Seminar Zufallsmatrizen series by
Jonas Jalowy
| Abstract: | This talk explores the limiting empirical zero distributions of polynomials as their degree tends to infinity. In particular, we discuss the role of the coefficients, the implications for (finite) free probability and how zeros evolve under certain differential flows. First, we shall focus on real-rooted polynomials and present a user-friendly approach to determine real limit zero distributions via the 'exponential profile' of the coefficients. Apart from applications to classical polynomial ensembles, this enables us to study the effect of repeated differentiation, the heat flow and finite free convolutions (which converges to the free convolutions even for non-compactly supported distributions). Second, we discuss random polynomials with i.i.d. rescaled coefficients and the evolution of their (complex) zeros under certain differential flows. For instance, the limiting zero distribution of Weyl polynomials undergoing the heat flow evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law. We interpret this from the perspective of free probability, optimal transport, and PDE's, and accompany the results by illustrative simulations. Based on joint works with Brian Hall, Ching-Wei Ho, Antonia Höfert, Zakhar Kabluchko, and Alexander Marynych. Within the CRC this talk is associated to the project(s): C6 |