Fluid limit and gelation in the frozen Erdős-Rényi random graph
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915134940643328 |
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| author | Haas, Bénédicte Viau, Vincent |
| author_facet | Haas, Bénédicte Viau, Vincent |
| contents | The frozen Erdős-Rényi random graph is a variant of the standard dynamical Erdős-Rényi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter $p\in [0,1]$ that quantifies the slowdown. At the time when all connected components of the graph have a (necessary unique) cycle, the graph is entirely frozen and the process stops. In this paper we study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. Our proofs are based on an adaption of Wormald's differential equation method. We also obtain, as a main application, a precise description of the asymptotic behavior of the first time when the graph is entirely frozen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fluid limit and gelation in the frozen Erdős-Rényi random graph Haas, Bénédicte Viau, Vincent Probability 05C80 60J75 60J25 The frozen Erdős-Rényi random graph is a variant of the standard dynamical Erdős-Rényi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter $p\in [0,1]$ that quantifies the slowdown. At the time when all connected components of the graph have a (necessary unique) cycle, the graph is entirely frozen and the process stops. In this paper we study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. Our proofs are based on an adaption of Wormald's differential equation method. We also obtain, as a main application, a precise description of the asymptotic behavior of the first time when the graph is entirely frozen. |
| title | Fluid limit and gelation in the frozen Erdős-Rényi random graph |
| topic | Probability 05C80 60J75 60J25 |
| url | https://arxiv.org/abs/2502.01424 |