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

Abstract

We show that the law of the $\gamma$-LQG metric (appropriately renormalized) is continuous in $\gamma \in (0,2)$ with respect to the local uniform topology of metrics on $\mathbf{C}\times \mathbf{C}$ whenever $\gamma$ lies on compact subsets of $(0,2)$. Moreover we show that as $\gamma \to 0$, the $\gamma$-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 $\xi >0$ (appropriately renormalized) is tight with respect to the topology on lower semicontinuous functions on $\mathbf{C}\times \mathbf{C}$ whenever $\xi$ 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 $\xi$. Finally we examine the limit as $\xi \to \infty$, which is a regime that has not been studied before. More precisely we show that if $D_h^{\xi }$ denotes the LQG metric with parameter $\xi >0$ (appropriately renormalized) associated with the whole-plane GFF $h$, the family of metrics $(D_h^{\xi }{)}^{1/\xi }$ is tight as $\xi \to \infty$ 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.