Lipschitz-type estimate for the frog model with Bernoulli initial configuration
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910389650849792 |
|---|---|
| author | Can, Van Hao Kubota, Naoki Nakajima, Shuta |
| author_facet | Can, Van Hao Kubota, Naoki Nakajima, Shuta |
| contents | We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters $p,q \in (0,1]$ is bounded from above and below by multiples of $|p-q|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18665 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lipschitz-type estimate for the frog model with Bernoulli initial configuration Can, Van Hao Kubota, Naoki Nakajima, Shuta Probability 60K35, 26A15, 82B43 We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters $p,q \in (0,1]$ is bounded from above and below by multiples of $|p-q|$. |
| title | Lipschitz-type estimate for the frog model with Bernoulli initial configuration |
| topic | Probability 60K35, 26A15, 82B43 |
| url | https://arxiv.org/abs/2403.18665 |