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On support sets of the critical Liouville Quantum Gravity
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abstract
We study fractal properties of support sets of the critical Liouville Quantum Gravity (cLQG) associated with the Gaussian Free Field in planar domains. Specifically, we completely characterize the gauge functions $\phi$ (subject to mild monotonicity conditions) for which the cLQG admits a support set of finite $\phi$-Hausdorff measure. As a corollary, we settle the conjecture that the cLQG is supported on a set of vanishing Hausdorff dimension. Our proofs are based on the fact that the cLQG describes the near-critical level sets of Discrete Gaussian Free Field.
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Cited by 1 Pith paper
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Extremal properties of the random walk local time
Expanded lecture notes proving that unvisited sites of a 2D simple random walk at sub-cover times converge to the thick points of the discrete Gaussian free field.
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