Continuity in $ξ$ and $ξ\to \infty$ tightness of the LQG metric
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| Format: | Preprint |
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2025
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| _version_ | 1866909797490622464 |
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| author | Kavvadias, Konstantinos |
| author_facet | Kavvadias, Konstantinos |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_15544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuity in $ξ$ and $ξ\to \infty$ tightness of the LQG metric Kavvadias, Konstantinos Probability 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. |
| title | Continuity in $ξ$ and $ξ\to \infty$ tightness of the LQG metric |
| topic | Probability |
| url | https://arxiv.org/abs/2509.15544 |