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

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.