## 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 âworst-caseâ 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