BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:William Turner (Imperial College London) DTSTART:20240405T130000Z DTEND:20240405T140000Z DTSTAMP:20260423T021651Z UID:ACPMS/38 DESCRIPTION:Title: F ree probability\, path developments and signature kernels as universal sca ling limits\nby William Turner (Imperial College London) as part of Al gebraic and Combinatorial Perspectives in the Mathematical Sciences\n\n\nA bstract\nScaling limits of random developments of a path into a matrix Lie Group have recently been used to construct signature-based kernels on pat h space\, while mitigating some of the dimensionality challenges that come with using signatures directly. Muça Cirone et al. have established a co nnection between the scaling limit of general linear group developments wi th Gaussian vector fields and the ordinary signature kernel\, while Lou et al. utilised unitary group developments and previous work of Chevyrev and Lyons to construct a path characteristic function distance. By leveraging the tools of random matrix theory and free probability theory\, we are ab le to provide a unified treatment of the limits in both settings under gen eral assumptions on the vector fields. For unitary developments\, we show that the limiting kernel is given by the contraction of a signature agains t 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.\n LOCATION:https://researchseminars.org/talk/ACPMS/38/ END:VEVENT END:VCALENDAR