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On support sets of the critical Liouville Quantum Gravity

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arxiv 2406.02814 v1 pith:VXAWGTX3 submitted 2024-06-04 math.PR math-phmath.MP

classification math.PRmath-phmath.MP
keywords clqgsetssupportcriticalfieldfreegaussiangravity
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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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  1. Extremal properties of the random walk local time

    math.PR 2025-02 unverdicted novelty 2.0 of 10

    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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