Shape Theorem for the Contact Process in a Dynamical Random Environment
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912549155373056 |
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| author | Reitmeier, Michel Seiler, Marco |
| author_facet | Reitmeier, Michel Seiler, Marco |
| contents | We study the contact process in a dynamical random environment defined on the vertices and edges of a graph. For a broad class of processes, we establish an asymptotic shape theorem for the set H_t, which represents the vertices that have been infected up to time t. More precisely, we show that this asymptotic shape is characterized -- similar to the basic contact process -- by a cone spanned by a convex set U, provided certain growth conditions are satisfied. Notably, we find that the asymptotic shape is independent of the initial configuration of the environment. Furthermore, we verify the growth conditions for various types of random environments, such as the contact process on a dynamical graph or a system with switching vertex states, where the monotonicity of the entire process is not guaranteed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16330 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape Theorem for the Contact Process in a Dynamical Random Environment Reitmeier, Michel Seiler, Marco Probability Primary 60K35, Secondary 82C22, 92D30 We study the contact process in a dynamical random environment defined on the vertices and edges of a graph. For a broad class of processes, we establish an asymptotic shape theorem for the set H_t, which represents the vertices that have been infected up to time t. More precisely, we show that this asymptotic shape is characterized -- similar to the basic contact process -- by a cone spanned by a convex set U, provided certain growth conditions are satisfied. Notably, we find that the asymptotic shape is independent of the initial configuration of the environment. Furthermore, we verify the growth conditions for various types of random environments, such as the contact process on a dynamical graph or a system with switching vertex states, where the monotonicity of the entire process is not guaranteed. |
| title | Shape Theorem for the Contact Process in a Dynamical Random Environment |
| topic | Probability Primary 60K35, Secondary 82C22, 92D30 |
| url | https://arxiv.org/abs/2508.16330 |