Wednesday, January 16, 2013

1301.2876 (Christophe Garban et al.)

Liouville Brownian motion    [PDF]

Christophe Garban, Rémi Rhodes, Vincent Vargas
We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or in the sphere in a conformal manner. Our construction works for all universality classes of planar maps satisfying $\gamma <\gamma_c=2$. In particular, this includes the interesting case of $\gamma=\sqrt{8/3}$ which corresponds to the conjectured scaling limit of large uniform planar $p$-angulations (with fixed $p\geq 3$). We start by constructing our process from some fixed point $x\in \R^2$ (or $x\in \S^2$). This amounts to changing the speed of a standard two-dimensional brownian motion $B_t$ depending on the local behaviour of the Liouville measure "$M_\gamma(dz) = e^{\gamma X} dz$" (where $X$ is a Gaussien Free Field, say on $\S^2$). A significant part of the paper focuses on extending this construction simultaneously to all points $x\in \R^2$ or $\S^2$ in such a way that one obtains a semi-group $P_\t$ (the Liouville semi-group). We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for $\gamma<\sqrt{2}$, the Liouville measure $M_\gamma$ is invariant under $P_\t$ (which in some sense shows that it is the right quantum gravity diffusion to consider). This Liouville Brownian motion enables us to give sense to part of the celebrated Feynman path integrals which are at the root of Liouville quantum gravity, the Liouville Brownian ones. Finally we believe that this work sheds some new light on the difficult problem of constructing a quantum metric out of the exponential of a Gaussian Free Field (see conjecture \ref{c.metric}).
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