Free probability, path developments and signature kernels as universal scaling limits
File(s)
Accepted version
Author(s)
Cass, Thomas
Turner, William
Type
Journal Article
Abstract
Random developments of a path into a matrix Lie group GN have recently been used to construct signature-based kernels on path space. Two examples include developments into GL(N ; R) and U (N ; C), the general
linear and unitary groups of dimension N . For the former, Muça Cirone et al. showed that the signature kernel is obtained via a scaling limit of developments with Gaussian vector fields. The second instance was used by Lou et al. 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.
linear and unitary groups of dimension N . For the former, Muça Cirone et al. showed that the signature kernel is obtained via a scaling limit of developments with Gaussian vector fields. The second instance was used by Lou et al. 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.
Date Issued
2026-04-01
Date Acceptance
2025-09-09
Citation
Annals of Applied Probability, 2026, 36 (2), pp.1082-1109
ISSN
1050-5164
Publisher
Institute of Mathematical Statistics
Start Page
1082
End Page
1109
Journal / Book Title
Annals of Applied Probability
Volume
36
Issue
2
Copyright Statement
Copyright © Institute of Mathematical Statistics, 2026. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2026-04-01