Bounds on the distance exponent for higher-dimensional Liouville first passage percolation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913790697668608 |
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| author | Hip, Andres A. Contreras Zhuang, Zijie |
| author_facet | Hip, Andres A. Contreras Zhuang, Zijie |
| contents | For $ξ\geq 0$ and $d \geq 3$, the higher-dimensional Liouville first passage percolation (LFPP) is a random metric on $ε\mathbb{Z}^d$ obtained by reweighting each vertex by $e^{ξh_ε(x)}$, where $h_ε(x)$ is a continuous mollification of the whole-space log-correlated Gaussian field. This metric generalizes the two-dimensional LFPP, which is related to Liouville quantum gravity. We derive several estimates for the set-to-set distance exponent of this metric, including upper and lower bounds and bounds on its derivative with respect to $ξ$. In the subcritical region for $ξ$, we derive estimates for the fractal dimension and show that it is continuous and strictly increasing with respect to $ξ$. In particular, our result is an important step towards proving a technical assumption made in previous work by the first author and Gwynne. These are also the first bounds on the distance exponent for LFPP in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on the distance exponent for higher-dimensional Liouville first passage percolation Hip, Andres A. Contreras Zhuang, Zijie Probability For $ξ\geq 0$ and $d \geq 3$, the higher-dimensional Liouville first passage percolation (LFPP) is a random metric on $ε\mathbb{Z}^d$ obtained by reweighting each vertex by $e^{ξh_ε(x)}$, where $h_ε(x)$ is a continuous mollification of the whole-space log-correlated Gaussian field. This metric generalizes the two-dimensional LFPP, which is related to Liouville quantum gravity. We derive several estimates for the set-to-set distance exponent of this metric, including upper and lower bounds and bounds on its derivative with respect to $ξ$. In the subcritical region for $ξ$, we derive estimates for the fractal dimension and show that it is continuous and strictly increasing with respect to $ξ$. In particular, our result is an important step towards proving a technical assumption made in previous work by the first author and Gwynne. These are also the first bounds on the distance exponent for LFPP in higher dimensions. |
| title | Bounds on the distance exponent for higher-dimensional Liouville first passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2504.09141 |