Convergence of Variants of Liouville First Passage Percolation and the Strong Liouville Quantum Gravity Coordinate Change Formula

This dissertation gathers two articles on Liouville quantum gravity (LQG), a theory of random two-dimensional geometry. Chapter 2 shows that Liouville first passage percolation (LFPP), the regularization used to construct the LQG metric, converges almost surely in the subcritical phase. This implies that the LQG metric is invariant under all spatial scalings simultaneously, and provides an improvement to the state-of-the-art bounds for the LFPP scaling constants. Chapter 3, which builds on material from Chapter 2, shows that the LQG metric coordinate change formula holds for all conformal maps simultaneously. Together with the analogous result for the LQG area measure, this formalizes the idea that LQG surfaces can be thought of as random equivalence classes of isometric surfaces.