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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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KPZ formulas for the Liouville quantum gravity metric
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by Ewain Gwynne and Joshua Pfeffer PDF
Trans. Amer. Math. Soc. 375 (2022), 8297-8324 Request permission

Abstract:

Let $\gamma \in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $\gamma$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \subset \mathbb {C}$ which is independent from $h$, the Hausdorff dimensions of $X$ with respect to the Euclidean metric and with respect to the $\gamma$-LQG metric $D_h$ are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum $\gamma$-LQG metric is equal to the exponent $d_\gamma > 2$ studied by Ding and Gwynne (2018), which describes distances in discrete approximations of $\gamma$-LQG such as random planar maps.

We also derive “worst-case” bounds relating the Euclidean and $\gamma$-LQG dimensions of $X$ when $X$ and $h$ are not necessarily independent, which answers a question posed by Aru (2015). Using these bounds, we obtain an upper bound for the Euclidean Hausdorff dimension of a $\gamma$-LQG geodesic which equals $1.312\dots$ when $\gamma = \sqrt {8/3}$; and an upper bound of $1.9428\dots$ for the Euclidean Hausdorff dimension of a connected component of the boundary of a $\sqrt {8/3}$-LQG metric ball.

We use the axiomatic definition of the $\gamma$-LQG metric, so the paper can be understood by readers with minimal background knowledge beyond a basic level of familiarity with the Gaussian free field.

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Additional Information
  • Ewain Gwynne
  • Affiliation: Department of Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom
  • Joshua Pfeffer
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 1275545
  • Received by editor(s): July 25, 2019
  • Received by editor(s) in revised form: December 17, 2019, December 26, 2019, and December 27, 2019
  • Published electronically: September 29, 2022
  • Additional Notes: The first author was supported by a Herchel Smith fellowship and a Trinity College junior research fellowship
    The second author was partially supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8297-8324
  • MSC (2010): Primary 60D05; Secondary 60G60
  • DOI: https://doi.org/10.1090/tran/8085