Liouville Brownian motion at criticality

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 [27], we start by constructing the critical LBM from one fixed point x ∈ R (or x ∈ S), 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)e2X(x) dx (where X is a Gaussian Free Field, say on S). Extending this construction simultaneously to all points in R 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 [20, 21].