Supercritical Liouville quantum gravity and CLE$_4$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917948855156736 |
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| author | Ang, Morris Gwynne, Ewain |
| author_facet | Ang, Morris Gwynne, Ewain |
| contents | We establish the first relationship between Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which corresponds to central charge values $\mathbf c_{\mathrm L} \in (1,25)$ or equivalently to complex values of $γ$ with $|γ|=2$. More precisely, we introduce a canonical supercritical LQG surface with the topology of the disk. We then show that for each $\mathbf c_{\mathrm L} \in (1,25)$ there is a coupling of this LQG surface with a conformal loop ensemble with parameter $κ=4$ (CLE$_4$) wherein the LQG surfaces parametrized by the regions enclosed by the CLE$_4$ loops are conditionally independent supercritical LQG disks given their boundary lengths. In this coupling, the CLE$_4$ is neither determined by nor independent from the LQG. Guided by our coupling result, we exhibit a combinatorially natural family of loop-decorated random planar maps whose scaling limit we conjecture to be the supercritical LQG disk coupled to CLE$_4$. We include a substantial list of open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11832 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Supercritical Liouville quantum gravity and CLE$_4$ Ang, Morris Gwynne, Ewain Probability Mathematical Physics We establish the first relationship between Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which corresponds to central charge values $\mathbf c_{\mathrm L} \in (1,25)$ or equivalently to complex values of $γ$ with $|γ|=2$. More precisely, we introduce a canonical supercritical LQG surface with the topology of the disk. We then show that for each $\mathbf c_{\mathrm L} \in (1,25)$ there is a coupling of this LQG surface with a conformal loop ensemble with parameter $κ=4$ (CLE$_4$) wherein the LQG surfaces parametrized by the regions enclosed by the CLE$_4$ loops are conditionally independent supercritical LQG disks given their boundary lengths. In this coupling, the CLE$_4$ is neither determined by nor independent from the LQG. Guided by our coupling result, we exhibit a combinatorially natural family of loop-decorated random planar maps whose scaling limit we conjecture to be the supercritical LQG disk coupled to CLE$_4$. We include a substantial list of open problems. |
| title | Supercritical Liouville quantum gravity and CLE$_4$ |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2308.11832 |