Concentration and central limit theorem for the averaging process on $\mathbb{Z}^{d}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913296910647296 |
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| author | Eide, Austin |
| author_facet | Eide, Austin |
| contents | In the averaging process on a graph $G = (V, E)$, a random mass distribution $η$ on $V$ is repeatedly updated via transformations of the form $η_{v}, η_{w} \mapsto (η_{v} + η_{w})/2$, with updates made according to independent Poisson clocks associated to the edge set $E$. We study the averaging process when $G$ is the integer lattice $\mathbb{Z}^{d}$. We prove that the process has tight asymptotic concentration around its mean in the $\ell^{1}$ and $\ell^{2}$ norms and use this to prove a central limit theorem. Previous work by Nagahata and Yoshida implies the central limit theorem when $d \geq 3$. Our results extend this to hold for all $d \geq 1$, and our techniques are likely applicable to other processes for which previously only the $d \geq 3$ case was tractable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_02351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concentration and central limit theorem for the averaging process on $\mathbb{Z}^{d}$ Eide, Austin Probability Combinatorics 60K35 In the averaging process on a graph $G = (V, E)$, a random mass distribution $η$ on $V$ is repeatedly updated via transformations of the form $η_{v}, η_{w} \mapsto (η_{v} + η_{w})/2$, with updates made according to independent Poisson clocks associated to the edge set $E$. We study the averaging process when $G$ is the integer lattice $\mathbb{Z}^{d}$. We prove that the process has tight asymptotic concentration around its mean in the $\ell^{1}$ and $\ell^{2}$ norms and use this to prove a central limit theorem. Previous work by Nagahata and Yoshida implies the central limit theorem when $d \geq 3$. Our results extend this to hold for all $d \geq 1$, and our techniques are likely applicable to other processes for which previously only the $d \geq 3$ case was tractable. |
| title | Concentration and central limit theorem for the averaging process on $\mathbb{Z}^{d}$ |
| topic | Probability Combinatorics 60K35 |
| url | https://arxiv.org/abs/2404.02351 |