The critical curve of long-range percolation on oriented trees
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909956911923200 |
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| author | Couronné, Olivier Gallo, Sandro Rolla, Leonardo T. |
| author_facet | Couronné, Olivier Gallo, Sandro Rolla, Leonardo T. |
| contents | We consider a long-range percolation model on homogeneous oriented trees with several lengths. We obtain the critical surface as the set of zeros of a specific polynomial with coefficients depending explicitly on the lengths and the degree of the tree. Restricting to the case of two lengths, we obtain new bounds on the critical parameters, monotonicity properties, as well as continuity of the critical curve, plus some partial results concerning its convexity. Our proofs rely on the study of the properties of the characteristic polynomial of the transition matrix of a multi-step Markov chain related to the model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The critical curve of long-range percolation on oriented trees Couronné, Olivier Gallo, Sandro Rolla, Leonardo T. Probability 60K35, 60J10 We consider a long-range percolation model on homogeneous oriented trees with several lengths. We obtain the critical surface as the set of zeros of a specific polynomial with coefficients depending explicitly on the lengths and the degree of the tree. Restricting to the case of two lengths, we obtain new bounds on the critical parameters, monotonicity properties, as well as continuity of the critical curve, plus some partial results concerning its convexity. Our proofs rely on the study of the properties of the characteristic polynomial of the transition matrix of a multi-step Markov chain related to the model. |
| title | The critical curve of long-range percolation on oriented trees |
| topic | Probability 60K35, 60J10 |
| url | https://arxiv.org/abs/2510.03922 |