On some stochastic evolutions on Laguerre-Polya functions

We prove uniqueness in law for three stochastic evolutions on normalised Laguerre-Pólya functions arising as scaling limits of matrix characteristic polynomial dynamics. In particular, this implies that Ahn's universal Brownian Gibbs line ensemble arising from random matrix products has the law of the logarithms of the reciprocal zeros of the solution, unique in law, to a simple stochastic equation on entire functions started from e⁻ᶻ. The method of proof relies on some integrable structure of the problem in the special form of a new duality with finite-dimensional diffusions which is closely related to intertwinings.