Following up on the evolution of complex roots of random polynomials and their advertised connection to free probability, this talk aims to give a full picture of this connection in the case of real rooted polynomials. First, I will present a general user-friendly approach for determining the limiting zero distribution as the degree tends to infinity in terms of the exponential profile of the coefficients. Then, we will explore a wide range of examples, covering many classical families of polynomials as well as certain operations on polynomials, such as Hadamard product, repeated differentiation, the heat flow and finite free convolutions. To this end, I will give a short crash-course on (finite) free probability theory. Finally, we close the circle by viewing the results of last week's talk from the perspective of free probability.
Real rooted polynomials, exponential profiles and (finite) free probability
Real rooted polynomials, exponential profiles and (finite) free probability