Almost Sure Convergence of Liouville First Passage Percolation
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929208145477632 |
|---|---|
| author | VI, Charles Devlin |
| author_facet | VI, Charles Devlin |
| contents | Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the family of random distance functions (metrics) $(D_h^ε)_{ε> 0}$ on $\mathbb{C}$ obtained heuristically by integrating $e^{ξh}$ along paths, where $h$ is a variant of the Gaussian free field. There is a critical value $ξ_{\text{crit}} \approx 0.41$ such that for $ξ\in (0, ξ_{\text{crit}})$, appropriately rescaled LFPP converges in probability uniformly on compact subsets of $\mathbb{C}$ to a limiting metric $D_h$ on $γ$-Liouville quantum gravity with $γ= γ(ξ) \in (0,2)$. We show that the convergence is almost sure, giving an affirmative answer to a question posed by Gwynne and Miller (2019). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08001 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Almost Sure Convergence of Liouville First Passage Percolation VI, Charles Devlin Probability Mathematical Physics 60D05 Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the family of random distance functions (metrics) $(D_h^ε)_{ε> 0}$ on $\mathbb{C}$ obtained heuristically by integrating $e^{ξh}$ along paths, where $h$ is a variant of the Gaussian free field. There is a critical value $ξ_{\text{crit}} \approx 0.41$ such that for $ξ\in (0, ξ_{\text{crit}})$, appropriately rescaled LFPP converges in probability uniformly on compact subsets of $\mathbb{C}$ to a limiting metric $D_h$ on $γ$-Liouville quantum gravity with $γ= γ(ξ) \in (0,2)$. We show that the convergence is almost sure, giving an affirmative answer to a question posed by Gwynne and Miller (2019). |
| title | Almost Sure Convergence of Liouville First Passage Percolation |
| topic | Probability Mathematical Physics 60D05 |
| url | https://arxiv.org/abs/2309.08001 |