Liouville Brownian Motion at Criticality

Abstract

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 Garban et al. (2013), we start by constructing the critical LBM from one fixed point $x\mathbb {R}^{2}$ (or $x\mathbb {S}^{2}$), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure M ^′(d x)=−X(x)e ^2X(x) d x (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 ^′. 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 Duplantier et al. (Ann. Probab. 42(5), 1769–1808, 2014), Duplantier et al. (Commun. Math. Phys. 330, 283–330 2014) and also establish new capacity estimates for the critical Gaussian multiplicative chaos.