Long-range one-dimensional internal diffusion-limited aggregation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917328791273472 |
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| author | da Costa, Conrado Thacker, Debleena Wade, Andrew |
| author_facet | da Costa, Conrado Thacker, Debleena Wade, Andrew |
| contents | We study internal diffusion limited aggregation on $\mathbb{Z}$, where a cluster is grown incrementally by adding, for each random walk dispatched from the origin, the first site it reaches outside the cluster. We assume that the increment distribution $X$ of the driving random walks has $\mathbb{E} X =0$, but need neither be simple nor symmetric, and can have $\mathbb{E} (X^2) = \infty$, for example. For the case where $\mathbb{E} (X^2) < \infty$, we prove that after $m$ of the random walks have been dispatched, all but $o(m)$ sites in the cluster form an approximately symmetric contiguous block around the origin. This strengthens a result of Blachère, for centred random walks whose increments have finite $3$rd moments, to the optimal moments condition. On the other hand, if $X$ is in the domain of attraction of a symmetric $α$-stable law, $1 < α<2$, we prove that the cluster contains a contiguous block of $δm +o(m)$ sites, where $0 < δ< 1$, but, unlike the finite-variance case, one may not take $δ=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10113 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-range one-dimensional internal diffusion-limited aggregation da Costa, Conrado Thacker, Debleena Wade, Andrew Probability 60K35 (Primary) 60F15, 60G50, 60K05, 82C24 (Secondary) We study internal diffusion limited aggregation on $\mathbb{Z}$, where a cluster is grown incrementally by adding, for each random walk dispatched from the origin, the first site it reaches outside the cluster. We assume that the increment distribution $X$ of the driving random walks has $\mathbb{E} X =0$, but need neither be simple nor symmetric, and can have $\mathbb{E} (X^2) = \infty$, for example. For the case where $\mathbb{E} (X^2) < \infty$, we prove that after $m$ of the random walks have been dispatched, all but $o(m)$ sites in the cluster form an approximately symmetric contiguous block around the origin. This strengthens a result of Blachère, for centred random walks whose increments have finite $3$rd moments, to the optimal moments condition. On the other hand, if $X$ is in the domain of attraction of a symmetric $α$-stable law, $1 < α<2$, we prove that the cluster contains a contiguous block of $δm +o(m)$ sites, where $0 < δ< 1$, but, unlike the finite-variance case, one may not take $δ=1$. |
| title | Long-range one-dimensional internal diffusion-limited aggregation |
| topic | Probability 60K35 (Primary) 60F15, 60G50, 60K05, 82C24 (Secondary) |
| url | https://arxiv.org/abs/2411.10113 |