Speaker
Jalowy, Paderborn University
Abstract
In this talk, I will first present a tool for zero distributions of polynomials, with a particular focus on its implications for characteristic polynomials. It turns out that the limiting zero distribution of real-rooted polynomials is explictly linked to the asymptotics of the coefficients in terms of an ‘exponential profile’. Apart from applications to classical polynomial ensembles, this enables us for instance to identify precise coefficient-asymptotics of the characteristic polynomials of sample covariance matrices matrices.
We also discuss implications on the zeros of many polynomial operations, such as the finite free convolution (given by the characteristic polynomials of asymptotically free sums of finite-dimensional matrices) and on the evolution of zeros when differential operators are applied to the (characteristic) polynomial. Most interestingly, the heat flow evolves the zeros of a polynomial into the free convolution with the semicircle law. Lastly, we will revisit the well-known hermiticity-evolution of the circular law into the elliptic and semicircle law, reappearing as zero distribution of a particular choice of complex rooted polynomials undergoing the heat flow. Ultimately, this leads to the (open) heat flow conjecture for characteristic polynomials, universality questions and to fascinating line-forming-phenomena of the zeros.
The results are based on joint works with Brian Hall, Ching Wei Ho, Antonia Höfert, Zakhar Kabluchko, Alexander Marynych.