Almost sure convergence of Liouville first passage percolation

Liouville first passage percolation (LFPP) with parameter \(\xi > 0\) $\xi >0$ is the family of random distance functions (metrics) \((D_h^{\epsilon })_{\epsilon > 0}\) ${(D_h^{ϵ})}_{ϵ>0}$ on \(\mathbb {C}\) $\mathbb{C}$ obtained heuristically by integrating \(e^{\xi h}\) $e^{\xi h}$ along paths, where h is a variant of the Gaussian free field. There is a critical value \(\xi _{\text {crit}} \approx 0.41\) ${\xi }_{\text{crit}}\approx 0.41$ such that for \(\xi \in (0, \xi _{\text {crit}})\) $\xi \in (0,{\xi }_{\text{crit}})$ , appropriately rescaled LFPP converges in probability uniformly on compact subsets of \(\mathbb {C}\) $\mathbb{C}$ to a limiting metric \(D_h\) $D_h$ on \(\gamma \) $\gamma$ -Liouville quantum gravity with \(\gamma = \gamma (\xi ) \in (0,2)\) $\gamma =\gamma (\xi )\in (0,2)$ . We show that the convergence is almost sure, giving an affirmative answer to a question posed by Gwynne and Miller (Invent. Math. 223, 213–333 (2021) [math.PR]).