Understand
In this paper, we construct the Brownian motion of Liouville Quantum Gravity with central charge $c=1$ (more precisely we restrict to the corresponding free field theory).
- Liouville quantum gravity with $c=1$ corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a $O(n=2)$ loop model or a $Q=4$-state Potts model embedded in a two dimensional surface in a conformal manner.
- Following \cite{GRV1}, we start by constructing the critical LBM from one fixed point $x\in\mathbb{R}^2$ (or $x\in\S^2$), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure $M'(dx)=-X(x)e^{2X(x)}\,dx$ (where $X$ is a Gaussian Free Field, say on $\mathbb{S}^2$).
- Extending this construction simultaneously to all points in $\mathbb{R}^2$ requires a fine analysis of the potential properties of the measure $M'$.