P\'olya--Schur problems and free probability

Abstract

In this work, we build a bridge between the P\'olya--Schur program and Voiculescu's free probability theory. A cornerstone of the former is the P\'olya--Benz Theorem, classifying a central family of real-root preserving operators on the space of polynomials, as those given by $f(\partial_z)$ for a Laguerre--P\'olya function $f$ and the derivative operator $\partial_{z}$. We prove that any free (additive) infinitely divisible distribution can be attained as the weak limit of root distributions of Appell polynomials $f_n(\partial_z)z^n$ as $n\to\infty$, for a suitably chosen sequence $f_n$ of Laguerre--P\'olya functions. Such questions on the (global) limiting distributions of real rooted polynomials belong to the active research area of finite free probability. In contrast to its standard tools, our approach allows for non-compactly supported limiting distributions, (barely) complex rooted polynomials and even provides the full microscopic description of the roots. Moreover, we extend our results to differential operators generating free multiplicative infinitely divisible distributions, to the rectangular free convolution, and to $f_n(\partial_z)p_n$ for real rooted polynomials $p_n$, implying a generalization of the recent connections between the heat flow and free Brownian motion to any free L\'evy process. As corollaries, we identify free stable distributions by choosing $f_n$ to be a fixed rescaled Laguerre--P\'olya function, and we prove various convergence results on the zero distributions of Jensen polynomials, e.g. the limiting root distribution of Jensen polynomial of the Riemann $\Xi$-function is given by the Cauchy distribution.