Abstract: Assiotis [Adv. Math., 410:Paper No. 108701, 28, 2022] gave an explicit bijection between (random or deterministic) functions in the Laguerre–Pólya class and ensembles of infinite random matrices whose distributions are invariant under unitary conjugation. An important feature of the Laguerre–Pólya class is that there is also an explicit bijection between it and linear operators on polynomials which both commute with the derivative operator and such that has only real roots for any polynomial which also has only real roots. Any such must be of the form . It turns out the roots of are approximately, and for the empirical measure in the large degree limit exactly, distributed according to a the law of a Lévy process starting from the roots of in Voiculescu’s free probability. This provides a analogue of the known connection between the backwards heat flow on polynomials and free Brownian motion for any free Lévy process. We will see explicitly how this free Lévy process depends , and which processes arise from important random matrix ensembles through the bijection of Assiotis. Based on joint work with Jonas Jalowy.