KPZ formulas for the Liouville quantum gravity metric

Let ? ? ( 0 , 2 stretchy="false">) \gamma \in (0,2) , let alttext="h"> h encoding="application/x-tex">h be the planar Gaussian free field, and alttext="upper D Subscript h"> D encoding="application/x-tex">D_h associated alttext="gamma"> encoding="application/x-tex">\gamma -Liouville quantum gravity (LQG) metric. We prove that for any random Borel set X subset-of double-struck upper C"> X ?<!-- ? <mml:mrow class="MJX-TeXAtom-ORD"> C encoding="application/x-tex">X \subset \mathbb {C} which is independent from Hausdorff dimensions of X"> encoding="application/x-tex">X with respect to Euclidean metric -LQG are a.s. related by (geometric) KPZ formula. As a corollary, we deduce dimension continuum equal exponent alttext="d gamma Baseline greater-than 2"> d &gt; encoding="application/x-tex">d_\gamma &gt; 2 studied Ding Gwynne (2018), describes distances in discrete approximations such as maps. also derive “worst-case” bounds relating when not necessarily independent, answers question posed Aru (2015). Using these bounds, obtain an bound geodesic equals alttext="1.312 ellipsis"> 1.312 …<!-- … encoding="application/x-tex">1.312\dots StartRoot 8 slash 3 EndRoot"> = 8 / 3 = \sqrt {8/3} ; alttext="1.9428 1.9428 encoding="application/x-tex">1.9428\dots connected component boundary alttext="StartRoot encoding="application/x-tex">\sqrt ball. use axiomatic definition metric, so paper can understood readers minimal background knowledge beyond basic level familiarity field.