Abstract
This paper is concerned with computing the spectral dimension of (critical) 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as derived by physicists (see Ambjørn et al. in JHEP 9802:010, 1998) 15 years ago.
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Communicated by Anton Bovier.
Partially supported by grant ANR-11-JCJC CHAMU.
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Rhodes, R., Vargas, V. Spectral Dimension of Liouville Quantum Gravity. Ann. Henri Poincaré 15, 2281–2298 (2014). https://doi.org/10.1007/s00023-013-0308-y
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DOI: https://doi.org/10.1007/s00023-013-0308-y