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

Abstract

We show that the $\gamma$-LQG metric (appropriately re-normalized) for $\gamma \in (0,2)$ 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 $\gamma$-LQG metric (appropriately re-normalized) is continuous in $\gamma \in [0,2]$ 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 $\mathbb{C}$.