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- W2947241649 abstract "Let $gammain (0,2)$, 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 $1.312dots$ when $gamma = sqrt{8/3}$; and an upper bound of $1.9428dots$ 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." @default.
- W2947241649 created "2019-06-07" @default.
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- W2947241649 date "2019-05-28" @default.
- W2947241649 modified "2023-09-27" @default.
- W2947241649 title "KPZ formulas for the Liouville quantum gravity metric" @default.
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