Liouville Brownian motion at criticality
Résumé
In this paper, we construct the Brownian motion of Liouville Quantum gravity when the underlying conformal field theory has a $c=1$ central charge. 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 $x\in\R^2$ (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 $\S^2$). Extending this construction simultaneously to all points in $\R^2$ requires a fine analysis of the potential properties of the measure $M'$. This allows us to construct a strong Markov process with continuous sample paths living on the support of $M'$, namely a dense set of Hausdorff dimension $0$. We finally construct the Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form of (critical) Liouville quantum gravity with a $c=1$ central charge. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in \cite{Rnew7,Rnew12}.
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