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The Critical Liouville Quantum Gravity Metric Induces the Euclidean Topology

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Abstract

We show that every possible metric associated with critical (γ = 2) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric. More precisely, we show that the optimal modulus of continuity of the critical LQG metric with respect to the Euclidean metric is a power of 1/log(1/∣·∣). Our result applies to every possible subsequential limit of critical Liouville first passage percolation, a natural approximation scheme for the LQG metric which was recently shown to be tight.

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Acknowledgements

We thank an anonymous referee for helpful comments on an earlier version of this article. We thank Jason Miller for helpful discussions. J.D. was partially supported by NSF grants DMS-1757479 and DMS-1953848. E.G. was partially supported by a Clay research fellowship.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jian Ding

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Ewain Gwynne

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  1. Jian Ding
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  2. Ewain Gwynne
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Correspondence to Jian Ding.

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Conflict of Interest J.D. is a member of Editorial Board of Frontiers of Mathematics and was not involved in the editorial review or the decision to publish this article. The authors declare no conflict of interest.

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Ding, J., Gwynne, E. The Critical Liouville Quantum Gravity Metric Induces the Euclidean Topology. Front. Math 19, 1–46 (2024). https://doi.org/10.1007/s11464-022-0106-2

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  • DOI: https://doi.org/10.1007/s11464-022-0106-2

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