  First passage percolation (FPP) on $\mathbb{Z}^d$ or $\mathbb{R}^d$ 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 $\gamma$-Liouville Quantum
Gravity (LQG) (where $\gamma \in (0,2)$ is a parameter) -- a random Riemannian
surface induced on the complex plane by the random metric tensor $e^{2\gamma
h/d_{\gamma}} ({dx^2+dy^2}),$ where $h$ is the whole plane, properly centered,
Gaussian Free Field (GFF), and $d_\gamma$ is the associated dimension. We
consider the unique infinite geodesic $\Gamma$ 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 $\Gamma$,
converge to deterministic measures on the space of generalized functions and
continuous metrics on the unit disk respectively. Moreover, we show that the
limiting objects living on the unit disk 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 KPZ class, the present work
initiates this research program in the context of LQG.
