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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.04431 |
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| _version_ | 1866917124412276736 |
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| author | Heeszel, Andrew |
| author_facet | Heeszel, Andrew |
| contents | The boundary modified contact process models an epidemic spreading in one dimension with two infection parameters, $λ_i$ and $λ_e$. Starting from a finite infected set, each edge of $\mathbb{Z}$ transmits the infection at rate $λ_i$ except for the rightmost and leftmost edges incident to infected vertices, which transmit the infection at rate $λ_e$. We show a strong law of large numbers and central limit theorem for the location of the rightmost infected vertex when $λ_i = λ_c$ and $λ_e = λ_c + \varepsilon$. We also show stretched exponential tail bounds in the fluctuations of the rightmost infected vertex, the extinction time of the process on the event of non-survival, and the probability of survival given the size of the initial infected region. Our results extend to the boundary modified contact process whenever $λ_c \leq λ_i < λ_e$, and solves an open problem first proposed by Andjel and Rolla in [1]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limiting Speed and Fluctuations for the Boundary Modified Contact Process Heeszel, Andrew Probability 60K35 The boundary modified contact process models an epidemic spreading in one dimension with two infection parameters, $λ_i$ and $λ_e$. Starting from a finite infected set, each edge of $\mathbb{Z}$ transmits the infection at rate $λ_i$ except for the rightmost and leftmost edges incident to infected vertices, which transmit the infection at rate $λ_e$. We show a strong law of large numbers and central limit theorem for the location of the rightmost infected vertex when $λ_i = λ_c$ and $λ_e = λ_c + \varepsilon$. We also show stretched exponential tail bounds in the fluctuations of the rightmost infected vertex, the extinction time of the process on the event of non-survival, and the probability of survival given the size of the initial infected region. Our results extend to the boundary modified contact process whenever $λ_c \leq λ_i < λ_e$, and solves an open problem first proposed by Andjel and Rolla in [1]. |
| title | Limiting Speed and Fluctuations for the Boundary Modified Contact Process |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2512.04431 |