Exact decay of the persistence probability in the Airy$_1$ process
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916393073508352 |
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| author | Ferrari, Patrik L. Liu, Min |
| author_facet | Ferrari, Patrik L. Liu, Min |
| contents | We consider the Airy$_1$ process, which is the limit process in KPZ growth models with flat and non-random initial conditions. We study the persistence probability, namely the probability that the process stays below a given threshold $c$ for a time span of length $L$. This is expected to decay as $e^{-κ(c) L}$. We determine an analytic expression for $κ(c)$ for all $c\geq 3/2$ starting with the continuum statistics formula for the persistence probability. As the formula is analytic only for $c>0$, we determine an analytic continuation of $κ(c)$ and numerically verify the validity for $c<0$ as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact decay of the persistence probability in the Airy$_1$ process Ferrari, Patrik L. Liu, Min Probability Mathematical Physics We consider the Airy$_1$ process, which is the limit process in KPZ growth models with flat and non-random initial conditions. We study the persistence probability, namely the probability that the process stays below a given threshold $c$ for a time span of length $L$. This is expected to decay as $e^{-κ(c) L}$. We determine an analytic expression for $κ(c)$ for all $c\geq 3/2$ starting with the continuum statistics formula for the persistence probability. As the formula is analytic only for $c>0$, we determine an analytic continuation of $κ(c)$ and numerically verify the validity for $c<0$ as well. |
| title | Exact decay of the persistence probability in the Airy$_1$ process |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2402.10661 |