Almost Sure Convergence of Liouville First Passage Percolation

Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the family of random distance functions (metrics) $(D_h^{\epsilon})_{\epsilon > 0}$ on $\mathbb{C}$ obtained heuristically by integrating $e^{\xi h}$ along paths, where $h$ is a variant of the Gaussian free field. There is a critical value $\xi_{\text{crit}} \approx 0.41$ such that for $\xi \in (0, \xi_{\text{crit}})$, appropriately rescaled LFPP converges in probability uniformly on compact subsets of $\mathbb{C}$ to a limiting metric $D_h$ on $\gamma$-Liouville quantum gravity with $\gamma = \gamma(\xi) \in (0,2)$. We show that the convergence is almost sure, giving an affirmative answer to a question posed by Gwynne and Miller (2019).