2013/11/22 by Rémi Rhodes, Vincent Vargas, Rhodes, Rémi +1 · 1 citation
Mathematics · Physics and Astronomy · #60J65 #81T40 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1311.5847
openalex publication_date 2013/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeGRV1, we start by constructing the critical LBM from one fixed point x∈ℝ2 (or x∈§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)e2X(x) dx (where X is a Gaussian Free Field, say on \mathbbS2). Extending this construction simultaneously to all points in ℝ2 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 associated Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in \citeRnew7,Rnew12 and also establish new capacity estimates for the critical Gaussian multiplicative chaos.