Free probability, path developments and signature kernels as universal scaling limits

Random developments of a path into a matrix Lie group $G_N$ have recently been used to construct signature-based kernels on path space. Two examples include developments into GL$(N;\mathbb{R})$ and $U(N;\mathbb{C})$, the general linear and unitary groups of dimension $N$. For the former, [MLS23] showed that the signature kernel is obtained via a scaling limit of developments with Gaussian vector fields. The second instance was used in [LLN23] to construct a metric between probability measures on path space. We present a unified treatment to obtaining large $N$ limits by leveraging the tools of free probability theory. An important conclusion is that the limiting kernels, while dependent on the choice of Lie group, are nonetheless universal limits with respect to how the development map is randomised. For unitary developments, 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. We provide a convergent numerical scheme for this equation, together with rates, which does not require computation of signatures themselves.