Continuity in $ξ$ and $ξ\to \infty$ tightness of the LQG metric

Abstract

We show that the law of the $γ$-LQG metric (appropriately renormalized) is continuous in $γ\in (0,2)$ with respect to the local uniform topology of metrics on $\mathbf{C} \times \mathbf{C}$ whenever $γ$ lies on compact subsets of $(0,2)$. Moreover we show that as $γ\to 0$, the $γ$-LQG metric (appropriately renormalized) converges to the Euclidean metric with respect to the local uniform topology of metrics on $\mathbf{C} \times \mathbf{C}$. More generally, we show that the law of the LQG metric with parameter $ξ>0$ (appropriately renormalized) is tight with respect to the topology on lower semicontinuous functions on $\mathbf{C} \times \mathbf{C}$ whenever $ξ$ lies on compact subsets of $(0,\infty)$, and any subsequential limit in law is non-trivial almost surely. If in addition we assume that the limit satisfies the triangle inequality almost surely, then it has the law of an LQG metric with an appropriate parameter $ξ$. Finally we examine the limit as $ξ\to \infty$, which is a regime that has not been studied before. More precisely we show that if $D_h^ξ$ denotes the LQG metric with parameter $ξ>0$ (appropriately renormalized) associated with the whole-plane GFF $h$, the family of metrics $(D_h^ξ)^{1 / ξ}$ is tight as $ξ\to \infty$ and any subsequential limit is non-trivial almost surely. If in addition we assume that the subsequential limit satisfies the triangle inequality almost surely, then the limit is a metric almost surely.