Exact decay of the persistence probability in the Airy$_1$ process

Patrik L. Ferrari; Min Liu

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arXiv · 2402.10661

Exact decay of the persistence probability in the Airy$_1$ process

Patrik L. Ferrari·Min Liu

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Abstract

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 $e^{-\kappa(c) L}$. We determine an analytic expression for $\kappa(c)$ 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 $\kappa(c)$ and numerically verify the validity for $c<0$ as well.

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BibTeX RIS

Patrik L. Ferrari, Min Liu. 2024-02-16. Exact decay of the persistence probability in the Airy$_1$ process. https://arxiv.org/abs/2402.10661

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