Convergence of the Subcritical $γ$-LQG Metric to the Critical 2-LQG Metric as $γ\to 2$

Abstract

We show that the $γ$-LQG metric (appropriately re-normalized) for $γ\in (0,2)$ converges in law to the critical 2-LQG metric (appropriately re-normalized) with respect to the local uniform topology of continuous functions on $\mathbb{C} \times \mathbb{C}$. Combining with the results in arXiv:2509.15544, we obtain that the law of the $γ$-LQG metric (appropriately re-normalized) is continuous in $γ\in [0,2]$ with respect to the local uniform topology of continuous functions on $\mathbb{C} \times \mathbb{C}$, where the 0-LQG metric corresponds to the Euclidean metric on $\mathbb{C}$.