Parabolic-hyperbolic dichotomy through half-plane coexistence
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915862158508032 |
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| author | Timár, Ádám |
| author_facet | Timár, Ádám |
| contents | Consider a unimodular random planar map (URM) with an invariant ergodic percolation having infinite primal and dual clusters. We say that there is half-plane coexistence if both the percolation and its dual have infinite clusters when restricted to a half-plane. Under mild assumptions on the percolation, we show that the URM is parabolic if and only if there is no half-plane coexistence, and it is hyperbolic if and only if there is half-plane coexistence. This extends the recent half-plane non-coexistence result for $\mathbb{Z}^2$ by Klausen and Kravitz and provides another manifestation of the parabolic-hyperbolic dichotomy for URM's. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parabolic-hyperbolic dichotomy through half-plane coexistence Timár, Ádám Probability Consider a unimodular random planar map (URM) with an invariant ergodic percolation having infinite primal and dual clusters. We say that there is half-plane coexistence if both the percolation and its dual have infinite clusters when restricted to a half-plane. Under mild assumptions on the percolation, we show that the URM is parabolic if and only if there is no half-plane coexistence, and it is hyperbolic if and only if there is half-plane coexistence. This extends the recent half-plane non-coexistence result for $\mathbb{Z}^2$ by Klausen and Kravitz and provides another manifestation of the parabolic-hyperbolic dichotomy for URM's. |
| title | Parabolic-hyperbolic dichotomy through half-plane coexistence |
| topic | Probability |
| url | https://arxiv.org/abs/2603.13642 |