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arXiv · 2309.08001

Almost Sure Convergence of Liouville First Passage Percolation

Charles Devlin VI
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Abstract

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

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Charles Devlin VI. 2024-01-12. Almost Sure Convergence of Liouville First Passage Percolation. https://arxiv.org/abs/2309.08001

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