Scaling limits of random developments of a path into a matrix Lie Group have recently been used to construct signature-based kernels on path space, while mitigating some of the dimensionality challenges that come with using signatures directly. General linear group developments have been shown to be connected to the ordinary signature kernel (Muça Cirone et al.), while unitary developments have been used to construct a path characteristic function distance (Lou et al.). By leveraging the tools of random matrix theory and free probability theory, we are able to provide a unified treatment of the limits in both settings under general assumptions on the vector fields. For unitary developments, we show that the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation. 

This is joint work with Thomas Cass.