Discrete snakes with globally centered displacements
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913945440223232 |
|---|---|
| author | Addario-Berry, Louigi Donderwinkel, Serte Goldschmidt, Christina Mitchell, Rivka |
| author_facet | Addario-Berry, Louigi Donderwinkel, Serte Goldschmidt, Christina Mitchell, Rivka |
| contents | We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaymé trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the Łukasiewicz path of a size-conditioned critical Bienaymé tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaymé tree and the height process of its associated looptree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete snakes with globally centered displacements Addario-Berry, Louigi Donderwinkel, Serte Goldschmidt, Christina Mitchell, Rivka Probability 60J80, 60F17, 60C05 We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaymé trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the Łukasiewicz path of a size-conditioned critical Bienaymé tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaymé tree and the height process of its associated looptree. |
| title | Discrete snakes with globally centered displacements |
| topic | Probability 60J80, 60F17, 60C05 |
| url | https://arxiv.org/abs/2505.21823 |