SLE$_κ(ρ)$ processes in the light cone regime on Liouville quantum gravity
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910728448901120 |
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| author | Kavvadias, Konstantinos Miller, Jason |
| author_facet | Kavvadias, Konstantinos Miller, Jason |
| contents | We study the relationship between certain SLE$_κ(ρ)$ processes, which are variants of the Schramm-Loewner evolution with parameter $κ$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $ρ> -2-κ/2$ and in this work we will focus on the light cone regime, meaning that $κ\in (0,4)$ and $\max(κ/2-4,-2-κ/2) < ρ< -2$. Such processes are self-intersecting even though ordinary SLE$_κ$ curves are simple for $κ\in (0,4)$. We show that such a process drawn on top of an independent $\sqrtκ$-LQG surface called a weight $(ρ+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $α$-stable Lévy process with $α= 1-2(ρ+2)/κ$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an SLE$_κ(κ-4)$ curve on an independent $\sqrtκ$-LQG surface for $κ\in (4/3,2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04005 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SLE$_κ(ρ)$ processes in the light cone regime on Liouville quantum gravity Kavvadias, Konstantinos Miller, Jason Probability We study the relationship between certain SLE$_κ(ρ)$ processes, which are variants of the Schramm-Loewner evolution with parameter $κ$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $ρ> -2-κ/2$ and in this work we will focus on the light cone regime, meaning that $κ\in (0,4)$ and $\max(κ/2-4,-2-κ/2) < ρ< -2$. Such processes are self-intersecting even though ordinary SLE$_κ$ curves are simple for $κ\in (0,4)$. We show that such a process drawn on top of an independent $\sqrtκ$-LQG surface called a weight $(ρ+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $α$-stable Lévy process with $α= 1-2(ρ+2)/κ$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an SLE$_κ(κ-4)$ curve on an independent $\sqrtκ$-LQG surface for $κ\in (4/3,2)$. |
| title | SLE$_κ(ρ)$ processes in the light cone regime on Liouville quantum gravity |
| topic | Probability |
| url | https://arxiv.org/abs/2412.04005 |