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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.05677 |
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| _version_ | 1866913345250000896 |
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| author | Archer, Eleanor Carrance, Ariane Ménard, Laurent |
| author_facet | Archer, Eleanor Carrance, Ariane Ménard, Laurent |
| contents | We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaymé-Galton-Watson trees with stably-decaying offspring tails with an exponent $α$ in (1, 2). We show that these quadrangulations admit subsequential scaling limits wich all have Hausdorff dimension $\frac{2α}{α-1}$ almost surely. We conjecture that the limits are unique and spherical, and we introduce a candidate for the limit that we call the $α$-stable sphere. In addition, we conduct a detailed study of volume fluctuations around typical points in the limiting maps, and show that the fluctuations share similar characteristics with those of stable trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable quadrangulations and stable spheres Archer, Eleanor Carrance, Ariane Ménard, Laurent Probability We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaymé-Galton-Watson trees with stably-decaying offspring tails with an exponent $α$ in (1, 2). We show that these quadrangulations admit subsequential scaling limits wich all have Hausdorff dimension $\frac{2α}{α-1}$ almost surely. We conjecture that the limits are unique and spherical, and we introduce a candidate for the limit that we call the $α$-stable sphere. In addition, we conduct a detailed study of volume fluctuations around typical points in the limiting maps, and show that the fluctuations share similar characteristics with those of stable trees. |
| title | Stable quadrangulations and stable spheres |
| topic | Probability |
| url | https://arxiv.org/abs/2405.05677 |