Date
2015
Dewey
Probabilités et mathématiques appliquées
Sujet
Gaussian multiplicative chaos; Critical Liouville quantum gravity; Brownian motion; Heat kernel; Potential theory
Journal issue
Potential Analysis
Volume
43
Number
2
Publication date
2015
Article pages
149-197
Publisher
Kluwer Academic Publishers
Author
Rhodes, Rémi
Vargas, Vincent
Type
Article accepté pour publication ou publié
Abstract (EN)
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 xR2 (or xS2), 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 S2). 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 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.