Diffusion in planar Liouville quantum gravity

Abstract

We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain D of the complex plane for which the Riemannian metric tensor at a point z ∈ D is given by exp(γh(z) − 1 2 γ 2 E(h(z) 2 )). Here h is an instance of the Gaussian Free Field on D and γ ∈ (0, 2) is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian Free Field, and show that it spends Lebesgue-almost all its time in the set of γ-thick points, almost surely.The diffusion is constructed by a limiting procedure after regularisation of the Gaussian Free Field. The proof is inspired by arguments of Duplantier-Sheffield for the convergence of the Liouville quantum gravity measure, previous work on multifractal random measures, and relies also on estimates on the occupation measure of planar Brownian motion by Dembo, Peres, Rosen and Zeitouni.A similar but deeper result has been independently and simultaneously proved

Heat Kernel for Liouville Brownian Motion and Liouville Graph Distance

“…In a recent work [21], upper and lower bounds have been obtained for a distance associated with the LQG (which is presumably related to Liouville graph distance considered in the present article), and for that distance the existence of the scaling exponent was established. From another perspective, the LBM has also drawn much interest recently, after it was constructed in [20,3]. In particular, the LBM heat kernel was constructed in [19], and on-diagonal bounds were derived in [32], implying that the spectral dimension of LBM equals 2.…”

“…where the limit exists almost surely due to [20,3]. It is not hard to check, using the a.s. convergence discussed in Section 2.3, that the limit in (32) does not depend on whether circle averages or white noise approximations are used.…”

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

“…• Various constructions using the so-called Liouville heat kernel, as defined in [GRV14], which is the heat kernel for Liouville Brownian motion [Ber15,GRV16a].…”

Liouville heat kernel: Regularity and bounds

“…Nonetheless, one can make sense of the volume form M γ associated to (1.1) by the theory of Gaussian multiplicative chaos [27]. Recently, the authors of [18] introduced the natural diffusion process (B t ) t 0 associated to (1.1), the so-called Liouville Brownian motion (LBM) (see also the work [7] for a construction of the LBM starting from one point) but also in [19] the associated heat kernel p γ t (x, y) (with respect to M γ ), called the Liouville heat kernel. One of the main motivations behind the introduction of the LBM and more specifically the Liouville heat kernel is to get an insight into the geometry of LQG: indeed, one can for instance note that there is a sizable physics literature in this direction (see the book [2] for a review).…”