The \(\varvec{d_\gamma /2}\)-Variation of Distance Profiles in \(\varvec{\gamma }\)-Liouville Quantum Gravity

Abstract

For Brownian surfaces with boundary and an interior marked point, a natural observable to consider is the distance profile, defined as the process of distances from the marked point to a variable point lying on the boundary. When the boundary is parametrized by the natural length measure on it, this distance profile turns out to be locally absolutely continuous to Brownian motion, and as a result, the boundary length measure itself has a natural interpretation as the quadratic variation process of the distance profile. In this paper, we extend this interpretation to \(\gamma \)-Liouville quantum gravity (\(\gamma \)-LQG), a one-parameter family of models of random geometry which is known to specialize to the case of Brownian geometry for the case \(\gamma =\sqrt{8/3}\). With \(d_\gamma \) denoting the Hausdorff dimension of \(\gamma \)-LQG, we show that for a \(\gamma \)-LQG surface with boundary, the natural boundary length measure can be interpreted (up to a constant factor) as the \(d_\gamma /2\)-variation process of the distance profile from an interior point.