{"doi":"10.48550/ARXIV.2203.01592","publisher":"arXiv","author":[{"first_name":"Viktor","full_name":"Bezborodov, Viktor","last_name":"Bezborodov"},{"last_name":"Di Persio","full_name":"Di Persio, Luca","first_name":"Luca"},{"first_name":"Peter","orcid":"0000-0002-8241-4076","id":"253182","full_name":"Kuchling, Peter","last_name":"Kuchling","orcid_put_code_url":"https://api.orcid.org/v2.0/0000-0002-8241-4076/work/168054692"}],"date_updated":"2024-09-23T07:50:48Z","language":[{"iso":"eng"}],"type":"journal_article","abstract":[{"text":"We consider the continuous-time frog model on $\\mathbb{Z}$. At time $t = 0$, there are $η(x)$ particles at $x\\in \\mathbb{Z}$, each of which is represented by a random variable. In particular, $(η(x))_{x \\in \\mathbb{Z} }$ is a collection of independent random variables with a common distribution $μ$, $μ(\\mathbb{Z}_+) = 1$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. Active particles perform a simple symmetric continuous-time random walk in $\\mathbb{Z} $ (that is, a random walk with $\\exp(1)$-distributed jump times and jumps $-1$ and $1$, each with probability $1/2$), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if $μ$ is the distribution of $e^{Y \\ln Y}$ with a non-negative random variable $Y$ satisfying $\\mathbb{E} Y &lt; \\infty$, then a.s. no explosion occurs. On the other hand, if $a \\in (0,1)$ and $μ$ is the distribution of $e^X$, where $\\mathbb{P} \\{X \\geq t \\} = t^{-a}$, $t \\geq 1$, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.","lang":"eng"}],"publication_status":"published","year":"2022","_id":"4949","status":"public","title":"Explosion and non-explosion for the continuous-time frog model","publication":"arXiv","date_created":"2024-09-20T08:35:05Z","user_id":"249224","citation":{"bibtex":"@article{Bezborodov_Di Persio_Kuchling_2022, title={Explosion and non-explosion for the continuous-time frog model}, DOI={<a href=\"https://doi.org/10.48550/ARXIV.2203.01592\">10.48550/ARXIV.2203.01592</a>}, journal={arXiv}, publisher={arXiv}, author={Bezborodov, Viktor and Di Persio, Luca and Kuchling, Peter}, year={2022} }","ama":"Bezborodov V, Di Persio L, Kuchling P. Explosion and non-explosion for the continuous-time frog model. <i>arXiv</i>. 2022. doi:<a href=\"https://doi.org/10.48550/ARXIV.2203.01592\">10.48550/ARXIV.2203.01592</a>","mla":"Bezborodov, Viktor, et al. “Explosion and Non-Explosion for the Continuous-Time Frog Model.” <i>ArXiv</i>, arXiv, 2022, doi:<a href=\"https://doi.org/10.48550/ARXIV.2203.01592\">10.48550/ARXIV.2203.01592</a>.","chicago":"Bezborodov, Viktor, Luca Di Persio, and Peter Kuchling. “Explosion and Non-Explosion for the Continuous-Time Frog Model.” <i>ArXiv</i>, 2022. <a href=\"https://doi.org/10.48550/ARXIV.2203.01592\">https://doi.org/10.48550/ARXIV.2203.01592</a>.","apa":"Bezborodov, V., Di Persio, L., &#38; Kuchling, P. (2022). Explosion and non-explosion for the continuous-time frog model. <i>ArXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2203.01592\">https://doi.org/10.48550/ARXIV.2203.01592</a>","ieee":"V. Bezborodov, L. Di Persio, and P. Kuchling, “Explosion and non-explosion for the continuous-time frog model,” <i>arXiv</i>, 2022.","short":"V. Bezborodov, L. Di Persio, P. Kuchling, ArXiv (2022).","alphadin":"<span style=\"font-variant:small-caps;\">Bezborodov, Viktor</span> ; <span style=\"font-variant:small-caps;\">Di Persio, Luca</span> ; <span style=\"font-variant:small-caps;\">Kuchling, Peter</span>: Explosion and non-explosion for the continuous-time frog model. In: <i>arXiv</i>, arXiv (2022)"}}