Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2605.10361 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918494201708544 |
|---|---|
| author | Beck-Tiefenbach, David Kaiser, Robin Überbacher, Julia |
| author_facet | Beck-Tiefenbach, David Kaiser, Robin Überbacher, Julia |
| contents | We investigate the limit shape of the single-source model for stochastic sandpiles on the integer line subject to $p$--topplings. In this model, an initial configuration of $n\in\mathbb{N}$ particles is placed at the origin and stabilized according to a random toppling rule depending on $p\in (0,1)$: an unstable vertex sends exactly one particle to its left neighbor with probability $p$, and independently sends exactly one particle to its right neighbor with probability $p$. We prove that as $n \to \infty$, the macroscopic limit shape of the final stable configuration is a symmetric interval around the origin. Furthermore, by analyzing the center of mass martingale, we establish a central limit theorem for the boundary fluctuations, showing that after proper rescaling, they converge to a Gaussian distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limit shape of single-source stochastic sandpiles with $p$-topplings on $\mathbb{Z}$ Beck-Tiefenbach, David Kaiser, Robin Überbacher, Julia Probability 60G42, 60F05, 60K35 We investigate the limit shape of the single-source model for stochastic sandpiles on the integer line subject to $p$--topplings. In this model, an initial configuration of $n\in\mathbb{N}$ particles is placed at the origin and stabilized according to a random toppling rule depending on $p\in (0,1)$: an unstable vertex sends exactly one particle to its left neighbor with probability $p$, and independently sends exactly one particle to its right neighbor with probability $p$. We prove that as $n \to \infty$, the macroscopic limit shape of the final stable configuration is a symmetric interval around the origin. Furthermore, by analyzing the center of mass martingale, we establish a central limit theorem for the boundary fluctuations, showing that after proper rescaling, they converge to a Gaussian distribution. |
| title | Limit shape of single-source stochastic sandpiles with $p$-topplings on $\mathbb{Z}$ |
| topic | Probability 60G42, 60F05, 60K35 |
| url | https://arxiv.org/abs/2605.10361 |