Almost Sure Convergence of Liouville First Passage Percolation

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).

Figures (2)

• Figure 1: Illustration of Definition \ref{['defn:AnnulusEvents']}. Condition 1 says that if the red path is a $D_h$-geodesic, then $\IfNoValueTF{-NoValue-} { \IfNoValueTF{-NoValue-} { \mathfrak{a}_{\epsilon}^{-1} \hat{D}_h^{\epsilon} } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_h^{-NoValue-} } } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_{-NoValue-}^{-NoValue-} } (u,v) \leq \frac{r \mathfrak{a}_{{\epsilon}}^{-1}}{r^{\xi Q} \mathfrak{a}_{{\epsilon}/r}^{-1}}(1 + \delta) D_h(u,v; \mathbb{A}_{r/2, 2r}(z))$, and analogously with $D_h$ and $\IfNoValueTF{-NoValue-} { \IfNoValueTF{-NoValue-} { \mathfrak{a}_{\epsilon}^{-1} \hat{D}_h^{\epsilon} } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_h^{-NoValue-} } } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_{-NoValue-}^{-NoValue-} }$ swapped. Condition 2 says that if the $D_h$-distance from $u$ to $v$ is larger than the $D_h$-distance from $u$ to the boundary of the gray annulus, then the red path cannot be a $\IfNoValueTF{-NoValue-} { \IfNoValueTF{-NoValue-} { \mathfrak{a}_{\epsilon}^{-1} \hat{D}_h^{\epsilon} } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_h^{-NoValue-} } } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_{-NoValue-}^{-NoValue-} }$-geodesic, and analogously with $D_h$ and $\IfNoValueTF{-NoValue-} { \IfNoValueTF{-NoValue-} { \mathfrak{a}_{\epsilon}^{-1} \hat{D}_h^{\epsilon} } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_h^{-NoValue-} } } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_{-NoValue-}^{-NoValue-} }$ swapped. Condition 3 says there is a path (shown in orange) around the cyan annulus with $D_h$-length at most $A$ times the $D_h$-distance across said annulus, and analogously with $D_h$ and $\IfNoValueTF{-NoValue-} { \IfNoValueTF{-NoValue-} { \mathfrak{a}_{\epsilon}^{-1} \hat{D}_h^{\epsilon} } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_h^{-NoValue-} } } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_{-NoValue-}^{-NoValue-} }$ swapped.
• Figure 2: Illustration of the proof of Proposition \ref{['prop:MainIterationArgument']}. Pictured is one of the annuli $\mathbb{A}_{\alpha r_{j-1}, r_{j-1}}(x_{j-1})$ such that $E_{r_{j-1}, \epsilon}(x_{j-1})$ occurs. The path in red is the $D_h$-geodesic $P$. Time $t_{j}$ is the first time after time $t_{j-1}$ that $P$ exits $B_{r_{j-1}}(x_{j-1})$, and $s_j$ is the last time before $t_j$ that $P$ leaves $B_{\alpha r_{j-1}}(x_{j-1})$. The segment $P|_{[s_j, t_j]}$ is a $D_h$-geodesic in $\mathbb{A}_{\alpha r_{j-1}, r_{j-1}}(x_{j-1})$, so condition 1 in Definition \ref{['defn:AnnulusEvents']} says $\IfNoValueTF{-NoValue-} { \IfNoValueTF{-NoValue-} { \mathfrak{a}_{\epsilon}^{-1} \hat{D}_h^{\epsilon} } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_h^{-NoValue-} } } { \mathfrak{a}_{-NoValue-}^{-1} \hat{D}_{-NoValue-}^{-NoValue-} } (P(s_j), P(t_j)) \leq \frac{r_{j-1} \mathfrak{a}_{\epsilon}^{-1}}{r_{j-1}^{\xi Q} \mathfrak{a}_{\epsilon/r_{j-1}}^{-1}} (1 + \delta) (t_j - s_j)$. The orange path is from condition 3 in Definition \ref{['defn:AnnulusEvents']}, and has $D_h$-length at most $A D_h(\partial B_{\alpha r_{j-1}}(x_{j-1}), \partial B_{r_{j-1}}(x_{j-1})) \leq A (t_j - s_j)$. Since $P$ crosses the orange path before time $t_{j-1}$ and after time $s_j$, it follows that $s_j - t_{j-1} \leq A(t_j - s_j)$. We use the latter to show that a positive proportion of $P$ is comprised of the segments $P|_{[s_j, t_j]}$.

Theorems & Definitions (39)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Proposition 3.1
• Definition 3.2