The scaling limit of planar maps with large faces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913827816210432 |
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| author | Curien, Nicolas Miermont, Grégory Riera, Armand |
| author_facet | Curien, Nicolas Miermont, Grégory Riera, Armand |
| contents | We prove that large Boltzmann stable planar maps of index $α\in (1;2)$ converge in the scaling limit towards a random compact metric space $\mathcal{S}_α$ that we construct explicitly. They form a one-parameter family of random continuous spaces ``with holes'' or ``faces'' different from the Brownian sphere. In the so-called dilute phase $α\in [3/2;2)$, the topology of $\mathcal{S}_α$ is that of the Sierpinski carpet, while in the dense phase $α\in (1;3/2)$ the ``faces'' of $\mathcal{S}_α$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or the behavior of geodesics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The scaling limit of planar maps with large faces Curien, Nicolas Miermont, Grégory Riera, Armand Probability 60D05, 60J65, 05C80, 60F17 We prove that large Boltzmann stable planar maps of index $α\in (1;2)$ converge in the scaling limit towards a random compact metric space $\mathcal{S}_α$ that we construct explicitly. They form a one-parameter family of random continuous spaces ``with holes'' or ``faces'' different from the Brownian sphere. In the so-called dilute phase $α\in [3/2;2)$, the topology of $\mathcal{S}_α$ is that of the Sierpinski carpet, while in the dense phase $α\in (1;3/2)$ the ``faces'' of $\mathcal{S}_α$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or the behavior of geodesics. |
| title | The scaling limit of planar maps with large faces |
| topic | Probability 60D05, 60J65, 05C80, 60F17 |
| url | https://arxiv.org/abs/2501.18566 |