Environment seen from infinite geodesics in Liouville Quantum Gravity

Abstract

First passage percolation (FPP) on ${\mathbb{Z}}^{\mathit{d}}$ or ${\mathbb{R}}^{\mathit{d}}$ is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesic and the typical one. We investigate this in the context of γ-Liouville quantum gravity ($\mathrm{LQG}$) (where $\mathit{\gamma }\in (0,2)$ is a parameter)—a random Riemannian surface induced on the complex plane by the random metric tensor ${\mathit{e}}^{2\mathit{\gamma }\mathit{h}/{\mathit{d}}_{\mathit{\gamma }}}(\mathit{d}{\mathit{x}}^2\mathbf{+}\mathit{d}{\mathit{y}}^2)$, where h is the whole plane, properly centered, Gaussian Free Field (GFF), and ${\mathit{d}}_{\mathit{\gamma }}$ is the associated dimension. We consider the unique infinite geodesic Γ from the origin, parametrized by the logarithm of its chemical length, and show that, for an almost sure realization of h, the distributions of the appropriately scaled field and the induced metric on a ball, rooted at a point “uniformly” sampled on Γ, converge to deterministic measures on the space of generalized functions and continuous metrics on the unit disk, respectively. Moreover, toward a better understanding of the limiting objects living on the unit disk, we show that they are singular with respect to their typical counterparts but become absolutely continuous away from the origin. Our arguments rely on unearthing a regeneration structure with fast decay of correlation in the geodesic owing to coalescence and the domain Markov property of the GFF. While there have been significant recent advances around this question for stochastic planar growth models in the Kardar–Parisi–Zhang universality class, the present work initiates this research program in the context of $\mathrm{LQG}$.