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Weyl's law in Liouville quantum gravity

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SSDW03 - Geometry, occupation fields, and scaling limits

Can you hear the shape of Liouville quantum gravity (LQG)?We obtain a Weyl law for the eigenvalues of Liouville Brownian motion:the number of eigenvalues in an interval grows linearly with the length of the interval. The proportionality constant is given by the Liouville measure of the domain and a certain deterministic constant which is computed explicitly and is, surprisingly, strictly greater than its Riemannian counterpart. This result (joint work with Mo Dick Wong) comes from a fine analysis of the LQG heat trace, which homogenises despite overwhelming pointwise fluctuations.We will also present a number of conjectures on the spectral geometry of LQG , notably suggesting a connection with “quantum chaos”.

This talk is part of the Isaac Newton Institute Seminar Series series.

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