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Can you hear the shape of Liouville quantum gravity (LQG)?
We obtain a Weyl law for the eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows linearly with n, with the proportionality
constant 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. After explaining this result and its context, as well as some related
estimates pertaining to the small-time behaviour of the heat kernel, I hope to also present a number of conjectures on the spectral geometry
of LQG.
These relate both to the behaviour of eigenfunctions (suggesting intriguing connections with so-called “quantum chaos”) and to that of
eigenvalues, for which we conjecture a connection to random matrix statistics.
This is joint work with Mo-Dick Wong (Durham).
