The Brownian loop measure on Riemann surfaces and applications to length spectra

The Brownian loop measure on the Riemann sphere was introduced by Lawler and Werner in studying the Schramm-Loewner evolution. We consider its generalization to an arbitrary Riemann surface and show that the lengths of closed geodesics are encoded in the Brownian loop measure. This gives a tool to study the length spectra of Riemann surfaces. In particular, using properties of the Brownian loop measure, we obtain a new identity between the length spectrum of a surface and that of the same surface with an arbitrary number of additional cusps. We also express the determinant of Laplacian of a compact hyperbolic surface as the total mass of Brownian loop measure renormalized according to the length spectrum. This is based on a joint work with Yuhao Xue (IHES).

• Speaker: Yilin Wang (IHES)
• Tuesday 03 December 2024, 14:00–15:00
• Venue: MR12.
• Series: Probability; organiser: ww295.