Liouville Brownian motion at criticality

TL;DR

This paper constructs the Liouville Brownian motion at criticality for c=1 Liouville Quantum Gravity, analyzing its properties and associated mathematical objects, and extends the theory of critical Gaussian multiplicative chaos.

Contribution

It provides a detailed construction of the critical Liouville Brownian motion and extends the theory of critical Gaussian multiplicative chaos with new capacity estimates.

Findings

Constructed the critical Liouville Brownian motion from a fixed point.

Extended the construction to all points, resulting in a Markov process on a fractal support.

Established new capacity estimates for critical Gaussian multiplicative chaos.

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 \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 $M'$. 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 \cite{Rnew7,Rnew12} and also establish new capacity estimates for the critical Gaussian multiplicative chaos.