Continuity in $ξ$ and ξ→ ∞ tightness of the LQG metric

We show that the law of the $γ$-LQG metric (appropriately renormalized) is continuous in γ∈ (0,2) with respect to the local uniform topology of metrics on C × C whenever $γ$ lies on compact subsets of $(0,2)$. Moreover we show that as γ→ 0, the $γ$-LQG metric (appropriately renormalized) converges to the Euclidean metric with respect to the local uniform topology of metrics on C × 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 C × C whenever $ξ$ lies on compact subsets of (0,∞), 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→ ∞, which is a regime that has not been studied before. More precisely we show that if Dₕ^ξ denotes the LQG metric with parameter $ξ>0$ (appropriately renormalized) associated with the whole-plane GFF h, the family of metrics (Dₕ^ξ)1 / ξ is tight as ξ→ ∞ 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.