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

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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).

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