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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.03996 |
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| _version_ | 1866913928117747712 |
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| author | Ding, Jian Gwynne, Ewain Zhuang, Zijie |
| author_facet | Ding, Jian Gwynne, Ewain Zhuang, Zijie |
| contents | We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random functions which approximates a log-correlated Gaussian field on $\mathbb R^d$. Consider the family of random metrics on $\mathbb R^d$ obtained by weighting the lengths of paths by $e^{ξh_n}$, where $ξ> 0$ is a parameter. We prove that if $ξ$ belongs to the subcritical phase (which is defined by the condition that the distance exponent $Q(ξ)$ is greater than $\sqrt{2d}$), then after appropriate re-scaling, these metrics are tight and that every subsequential limit is a metric on $\mathbb R^d$ which induces the Euclidean topology. We include a substantial list of open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03996 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension Ding, Jian Gwynne, Ewain Zhuang, Zijie Probability Mathematical Physics We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random functions which approximates a log-correlated Gaussian field on $\mathbb R^d$. Consider the family of random metrics on $\mathbb R^d$ obtained by weighting the lengths of paths by $e^{ξh_n}$, where $ξ> 0$ is a parameter. We prove that if $ξ$ belongs to the subcritical phase (which is defined by the condition that the distance exponent $Q(ξ)$ is greater than $\sqrt{2d}$), then after appropriate re-scaling, these metrics are tight and that every subsequential limit is a metric on $\mathbb R^d$ which induces the Euclidean topology. We include a substantial list of open problems. |
| title | Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2310.03996 |