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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2511.15089 |
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| _version_ | 1866911497634971648 |
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| author | Dey, Partha S. Etesami, S. Rasoul Gopalan, Aditya S. |
| author_facet | Dey, Partha S. Etesami, S. Rasoul Gopalan, Aditya S. |
| contents | We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random. Co-located points are merged into a single point, and the resulting simple point process is rescaled to unit intensity. We show that, when the point processes are shifted so that there is a point at the origin, the dynamics have a unique weak limit when the initial point process is renewal. For this limiting point process, the gap distribution has exponential tails. We also show that for the time-reversed process and with an appropriate scaling in space, there is a limiting (random) distribution function on $\mathbb{R}$, whose associated measure assigns to $\mathbb{R}$ a measure corresponding to the gap between consecutive points. Finally, we discuss several relevant research directions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling Limit of a Stochastic Clustering Model on $\mathbb{R}$ Dey, Partha S. Etesami, S. Rasoul Gopalan, Aditya S. Probability We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random. Co-located points are merged into a single point, and the resulting simple point process is rescaled to unit intensity. We show that, when the point processes are shifted so that there is a point at the origin, the dynamics have a unique weak limit when the initial point process is renewal. For this limiting point process, the gap distribution has exponential tails. We also show that for the time-reversed process and with an appropriate scaling in space, there is a limiting (random) distribution function on $\mathbb{R}$, whose associated measure assigns to $\mathbb{R}$ a measure corresponding to the gap between consecutive points. Finally, we discuss several relevant research directions. |
| title | Scaling Limit of a Stochastic Clustering Model on $\mathbb{R}$ |
| topic | Probability |
| url | https://arxiv.org/abs/2511.15089 |