# KPZ formulas for the Liouville quantum gravity metric

## TL;DR

This paper establishes the KPZ relation between Euclidean and Liouville quantum gravity (LQG) dimensions, proving a key formula for the LQG metric and deriving bounds for geodesic and boundary dimensions, connecting discrete models to continuum theory.

## Contribution

It proves the KPZ formula for Hausdorff dimensions in LQG, relates discrete and continuum models, and provides bounds on geodesic and boundary dimensions in LQG.

## Key findings

- Hausdorff dimensions of independent sets follow KPZ formula almost surely.
- Dimension of the continuum $ frac{ ext{LQG metric}}{ ext{Euclidean}}$ is $d_ ext{ ext{LQG}} > 2$.
- Upper bounds for Euclidean dimensions of LQG geodesics and boundary components.

## 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.