Law of large numbers for activated random walk on villages
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909025310867456 |
|---|---|
| author | Ráth, Balázs Richey, Jacob Salánki, Miklós |
| author_facet | Ráth, Balázs Richey, Jacob Salánki, Miklós |
| contents | We consider activated random walk (ARW), an interacting particle system and prototypical model of self-organized criticality in a setting which combines mean-field behavior with the geometry of an arbitrary graph, which we call the village model of ARW, or VARW for short. VARW is obtained from a fixed graph by replacing each vertex with a 'village' that consists of n replicas of that vertex. We focus on VARW where particles walk according to a strictly sub-stochastic transition kernel on a finite underlying graph, so mass is sometimes lost (which guarantees that the system eventually stabilizes almost surely). Under a subcriticality assumption on the initial state we prove a law of large numbers as n goes to infinity for the resulting stable configuration of particles and the odometer of the process, to a limit which is uniquely characterized by a system of non-linear equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06887 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Law of large numbers for activated random walk on villages Ráth, Balázs Richey, Jacob Salánki, Miklós Probability We consider activated random walk (ARW), an interacting particle system and prototypical model of self-organized criticality in a setting which combines mean-field behavior with the geometry of an arbitrary graph, which we call the village model of ARW, or VARW for short. VARW is obtained from a fixed graph by replacing each vertex with a 'village' that consists of n replicas of that vertex. We focus on VARW where particles walk according to a strictly sub-stochastic transition kernel on a finite underlying graph, so mass is sometimes lost (which guarantees that the system eventually stabilizes almost surely). Under a subcriticality assumption on the initial state we prove a law of large numbers as n goes to infinity for the resulting stable configuration of particles and the odometer of the process, to a limit which is uniquely characterized by a system of non-linear equations. |
| title | Law of large numbers for activated random walk on villages |
| topic | Probability |
| url | https://arxiv.org/abs/2605.06887 |